A proposal of ion and aerosol A proposal of ion and aerosol vertical gradient measurement vertical gradient measurement
( ( as as an example of application an example of application of the heat transfer equations of the heat transfer equations ) )
H. Tammet
Pühajärve 2008
Measuring of vertical profiles by means of several simultaneously working instruments is expensive and requires
extra fine calibration of instruments.
Long tube 2
Short tube 4 Long tube 1
Long tube 3
Instrument
Inlet switch
Tower
How to estimate
the inlet losses?
Incropera, F.P. and Dewitt, D.P.:
Fundamentals of Heat and Mass Transfer, Fifth Edition, Wiley, New York, 2002),
see pages 355 357, 470, and 491 492. ‑ ‑
The mass transfer equations are derived from the heat transfer equations replacing
the Nusselt number with the Sherwood number and
the Prandtl number with the Schmidt number.
p – air pressure, Pa
T – absolute temperature, K
Φ – air flow rate through the tube, m3s–1 d – internal diameter of the tube, m
L – length of the tube, m
Z – electric mobility of ions or particles, m2 V–1 s–1
Diffusion coefficient of ions or particles
e D kTZ
Air density 3 kg m 3 K
Pa 10 :
49 .
3
T:
p
Air kinematic viscosity 2 1
5 1.8 m s
Pa :
K) : 10 (
5 .
5 -
p
T
Average linear speed of the air in the tube 4 2 u d
Dynamic pressure
2 u2
pd
Reynolds number
ud Re
.
Average time of the passage ,s u t L
Moody friction coefficient, Petukhov approximation
0.76lnRe1.64
2 f
valid if 3000 < Re < 5000000
Pressure drop along the long tube pd
d f L
p
Schmidt number Sc = ν / D
Sherwood number, Gnielinski approximation
) 1 Sc
( ) 8 / ( 7 . 12 1
Sc ) 1000 )(Re
8 /
Sh ( 1/2 2/3
f f
valid if 3000 < Re < 5000000 and 0.5 < Sc < 2000
Linear deposition velocity on the internal wall of the tube d Sh
h D
n – concentration of particles
N – flux of particles through a section of the long tube N = nΦ No – flux through the inlet of the long tube
Loss of particles in a short section of the tube (length dL) dN = –(πd×dL)×h×n = –πd×dL×h×N / Φ
Relative loss in a short section dNN dh dL
Relative pass exp( ) exp( Sh)
o
dhL DL
N N
.
The inlet parameters were p, T, Φ, d, L, Z
The outlet parameters are Re, Sc, t, p
d, Δp, N/N
o3 / 1 5
/
4 Pr
Re 023 .
0 Nu
:Relative loss in a long tube: 7/15 1/5 4/5
3 / 3 2
/ 2 5
/ 1
092 4 .
0 d
Z L e
kT
.
Sorry, the explicit equations derived according to the algorithm above are awkward.
An explicit equation can be derived using simplified approximation by Colburn
However, the Gnielinski approximation is
strongly preferred for quantitative calculations.
p = 1000 mb, T = 17 C
l/s m mm Z Re Sc t pd dp pass%
10 8 50 3.162 17110 1.9 1.6 15.6 75.1 20.0 10 8 50 1.000 17110 6.0 1.6 15.6 75.1 44.3 10 8 50 0.316 17110 18.8 1.6 15.6 75.1 67.4 10 8 50 0.100 17110 59.6 1.6 15.6 75.1 82.9 10 8 150 3.162 5703 1.9 14.1 0.2 0.4 54.3 10 8 150 1.000 5703 6.0 14.1 0.2 0.4 74.6 10 8 150 0.316 5703 18.8 14.1 0.2 0.4 87.1 10 8 150 0.100 5703 59.6 14.1 0.2 0.4 93.7 20 8 50 3.162 34221 1.9 0.8 62.4 252.1 24.0 20 8 50 1.000 34221 6.0 0.8 62.4 252.1 47.5 20 8 50 0.316 34221 18.8 0.8 62.4 252.1 69.3 20 8 50 0.100 34221 59.6 0.8 62.4 252.1 83.9 20 8 150 3.162 11407 1.9 7.1 0.8 1.4 56.6 20 8 150 1.000 11407 6.0 7.1 0.8 1.4 75.3 20 8 150 0.316 11407 18.8 7.1 0.8 1.4 87.3 20 8 150 0.100 11407 59.6 7.1 0.8 1.4 93.8
