## יום שישי, 17 במאי 2013

### STACK EFFECT - THERMALLY INDUCED FLOW

Wikipedia gives the equations with no explanation or proof. The references don't help much.
The driving force is buoyancy of the hot gas on cooler gas. In most materials, density reduces as temperature increase. Force is proportional to difference in density.
The pressure difference between lower side and upper side of the building or chimney is according to Bernoulli's principle, based on conversation of energy
P + ρ g h + ½ ρ v2 = constant
Where:
v = fluid flow speed
g = gravity acceleration
h = elevation above a reference plane
P = [[pressure]] at the chosen point
ρ = [[density]] of the fluid at all points in the fluid.

For low speed, pressure difference :

∆P = (ρa – ρ) ∆h

ρa = ambient density of fluid, outside the building
∆h = vertical distance

Assuming air to be ideal (or perfect) gas
ρ = P/ RT
R = gas constant
T = temperature
For pressure 1 to 5 Bar and temperatures 250 to 1000 Kelvin (-50 to 700 Celsius), quite far from critical point, the assumption is true within 1%
∆P = Pa g ∆h (1/To – 1/Ti) / R

Where :
Pa = Pressure outside
To = Absolute temperature outside (Kelvin)
Ti = Absolute temperature of gas inside the building / chimney (Kelvin)

Substituting value for Standard Atmosphere at ground level and gravitation:

∆P = 101,325 x 9.807 / 287 (1/To – 1/Ti) x ∆h = 3474 (1/To – 1/Ti) x ∆h

Flow rate can be evaluated, by manipulating the equation at White p.15
Q = Cd A (∆P/ρ)1/2

Cd = discharge coefficient
A = Cross section of gas flow = Internal diameter of chimney of opening of building

Q = C A (2 g ∆h (Ti- To)/Ti )1/2

∆h is height of chimney or the elevation above neutral plane (NLP)
Typical value for C is 0.6 to 0.7 

 Frank M. White, Fluid Mechanics, 6th Ed, McGraw Hill 2008, p.11
 2007 ASHRAE Handbook – HVAC Applications p. 52.2  uses 3460 instead of 3474
 Footnote 1
 Footnote 3