How is the concentration of particulate matter calculated per cubic meter
The data that you are having are called emission fluxes. If these data are available as global averages then one can calculate global mean concentration by multiplying the residence time of the species with emission flux.
However, at the city or grid level, it is not straightforward to calculate concentration from emission fluxes. In order to calculate, concentration, one requires information of rate of dispersion in three dimensions as well as species amount brought-in by the wind. The dispersion is generally estimated using atmospheric transport models. See for example Gadhavi et al. (2015) have estimated concentration from emission fluxes for black carbon particles.
Technically speaking, one can calculate concentration by first dividing the region where the concentration is needed in three dimensional grid-boxes and by calculating mass fluxes at each interface of the grid-boxes as well as chemical transformation within box. The chemical transformation for species like PM10 or PM2.5 may not be significant but it can be significant for species like SO2. The mass flux calculations will require knowledge on wind velocities across interfaces, diffusion mixing and turbulent mixing within the box as well as removal processes such as dry deposition and wet deposition.
Here, a confusing issue is the size of grid-boxes. A different size may lead to a different concentration number. Though, there is no hard and fast rule about size of grid-boxes, the decision often depends on resolution of available flux and meteorological data as well as end purpose. Suppose, one is looking for comparing satellite based columnar concentration, he/she may take grid-box height that covers entire atmosphere but if one is planning to compare a surface mass concentration measurement typically obtained by air pollution monitoring stations, vertical height of grid-box should be as small as possible preferably less than boundary layer height.
Most emission flux data or emission inventories are available for the emission near Earth's surface. They should be considered as emission flux across the bottom face of the lowest grid-box. The grid box that one but the lowest grid-box will have emission fluxes from top of the lowest grid-box dictated by meteorology and dispersion physics. And the gain of this grid-box will be losses (or negative flux) of the lowest grid box. Besides, losing mass to above, the lowest grid-boxes will also loose mass through process called dry deposition. If the grid-boxes covers the region with rain, then all the grid-boxes from cloud level to surface will also have wet deposition or wet removal. Same applies to mass fluxes in horizontal direction. The lowest grid-box gains mass from bottom face because of emissions but looses to neighboring boxes by horizontal wind. A balance between mass fluxes across all the faces will be concentration of given grid-box.
A crude estimation of surface concentration can be obtained by making a certain assumptions, like by taking grid-box height equal to mixing layer height and assuming horizontally uniform emission fluxes across large region. Under this assumptions one can calculate concentration by multiplying emission fluxes with characteristic time period say 24 hour and dividing it by volume of grid-box. The scientific rational behind this assumption is that mixing height is the height of the atmosphere were turbulence is very high and the loss process becomes dominant only after this volume is filled. Also, when the emission region is sufficiently large, horizontal movement of mass will cancel out across opposing face of the grid-box.
Reference
Gadhavi, H. S., Renuka, K., Ravi Kiran, V., Jayaraman, A., Stohl, A., Klimont, Z., and Beig, G.: Evaluation of black carbon emission inventories using a Lagrangian dispersion model – a case study over southern India, Atmos. Chem. Phys., 15, 1447–1461, https://doi.org/10.5194/acp-15-1447-2015, 2015.
Chapter 2 Atmospheric Trace Constituents in Atmospheric Chemistry and Physics by Seinfeld and Pandis, 2006.