How to make a design for smoke management in public areas(Standards ,codes, methods ,and formulas) ?

This is an old question

you can use ASHRAE Atrium. Calc. excel sheet and you will find a good description for the manual procedures 

2.1.1 Convective Heat Release Rate

The convective portion of the heat release rate (HRR) is

Q c = XQ (1)

where

Q c = convective heat release rate, Btu/s (kW),

Q = heat release rate, Btu/s (kW),

X= convective fraction, dimensionless.

The above equation is the same as the equation of NFPA 92 section 5.5.1.3. A convective fraction of 0.7 is commonly used for most design applications, but the value of X is much different for some fuels.

2.1.2 Axisymmetric Plume

Design fires located in the large atrium space are usually analyzed as having an axisymmetric plume (Figure 1a). For z greater than or equal to z l (z >= z l ), the mass flow of this kind of plume is

m=(0.071 Qc ^1/3 Z^5/3) + (0.0018 Qc)

For z less than z l (z < z l ), the mass flow is 

m = 0.032 Qc^3/5 Z

where

m = mass flow in the plume, lb/s (kg/s),

Q c = convective heat release rate, Btu/s (kW),

z = distance above the base of the fire, ft (m),

z l = limiting elevation, ft (m). 

For smoke temperature:

Ts = To + (KsQc / m Cp)

where:

Ts = smoke layer temperature

To = ambient or outdoor temperature

Ks = 1 for steady smoke exhaust

m = exhaust mass flow rate

Cp = spacific heat for smoke

for density of smoke:

ps = Patm / (R(Ts+273))

where

ps = smoke density kg/m3

Patm = atmospheric pressure Pa

R = gas constant = 287 j/kg k

Ts = smoke layer temperature

For smoke volume flow rate:

V = m / ps

all dimensions in SI units  

References

Klote, J. H., Milke, J. A., Turnbull, P. G., Kashef, A., Ferreira, M. J. 2012. Handbook of Smoke Control Engineering. Atlanta: ASHRAE.

NFPA. 2012. Standard for Smoke Control Systems NFPA 92. Quincy, MA: National Fire Protection Association.

ICC. 2012. International Building Code ® . Washington, D.C.: International Code Council. 

 the best way to understand & preform smoke calculation is Atrium Calculation ASHRAE