A proposal of ion and aerosol A proposal of ion and aerosol vertical gradient measurement vertical gradient measurement ((as as an example of applicationan example of applicationof the heat transfer equationsof the heat transfer equations))

H. Tammet

Pühajärve 2008

Measuring of vertical profiles by meansof several simultaneously working

instruments is expensive and requires extra fine calibration of instruments.

Long tube 2Short tube 4

Long tube 1

Long tube 3

Instrument

Inlet switch

Tower

How to estimatethe 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 particlesekTZ

D

Air density 33 mkg

KPa:

1049.3 T:p

Air kinematic viscosity 121.8

5 sm Pa:

K):(105.5 -

pT

Average linear speed of the air in the tube 2

4d

u

Dynamic pressure 2

2upd

Reynolds number

udRe

.

Average time of the passage s ,uL

t

Moody friction coefficient, Petukhov approximation

264.1Reln76.0 f

valid if 3000 < Re < 5000000

Pressure drop along the long tube dpdL

fp

Schmidt number Sc = ν / D

Sherwood number, Gnielinski approximation

)1Sc()8/(7.121Sc)1000)(Re8/(

Sh 3/22/1

ff

valid if 3000 < Re < 5000000 and 0.5 < Sc < 2000

Linear deposition velocity on the internal wall of the tube

ShdD

h

n – concentration of particlesN – 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 dLdh

NdN

Relative pass )Shexp()exp(o

DLdhLNN

.

The inlet parameters were p, T, Φ, d, L, Z

The outlet parameters are Re, Sc, t, pd, Δp, N/No

3/15/4 PrRe023.0Nu

:Relative loss in a long tube: 5/45/115/7

3/23/25/1

4092.0

dZL

ekT

.

Sorry, the explicit equations derived accordingto the algorithm above are awkward.

An explicit equation can be derived usingsimplified approximation by Colburn

However, the Gnielinski approximation isstrongly 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

Conclusion 1:

The turbulent adsorption of ions and nanometer particles in long inlet tubes can be easily and exactly estimated when using the heat transfer equations

Conclusion 2:

The turbulent adsorption is low enough to make possible the gradient measurements usingsingle instrument and commutative inlet tubes

Thank you !