Transcript Mathematical model EXCEL SHEET

Acknowledgement: Research is subsidized by
J04/98: 212200008
15 x 21
R.Žitný, J.Thýn
Department of Process Engineering
CTU in Prague, Faculty of Mechanical Engineering
T0,p0,Q
E-mail: [email protected]
Electrodes of heater are perforated - cross flow from lateral channels
should improve uniformity of temperatures in central channel.
However,
L
H
Warm liquid (Te) stands
in the right channel if
p1-p0=0gL[1-(Te-T0)]
Linear relationship between
pressure drop and flow-rate
through slits in electrodes
•Mass balances
•Heat transfer
•Momentum
•Tracer balances
Problem:
Control volumes are characterised by:
•parabolic velocity profile
•linear increase of temperature
•residence times - serie of mixers
Parallel flows are typical for many apparatuses of process industries, e.g. shell&tube or plate heat
exchangers, heaters, reactors. Sometimes instabilities or just only non-uniform distribution of
flow in parallel channels occur if the apparatus operates at non-isothermal conditions. Parallel
flow instabilities have been observed also in lateral channels of ohmic heater. One parallel stream
is delayed or even stopped if the temperature increase is too high. Cross-flow through perforation
of electrodes is also suppressed. These unpleasant effects are caused by natural convection.
Mathematical description
http://www.fsid.cvut.cz/en/u218/peoples/zitny/imagohm/instabil/instabil.doc
Theoretical analysis predicts two solutions of temperature and flow-fields:
 Symmetric solution (flow-rates and temperatures in lateral channels are equal)
 Asymmetric solution exists within a certain range of flow-rates and heating power.
Asymmetric solution can be interpreted as a magnitude of disturbance causing instability of flow.
Mathematical model enables to identify parameters having significant influence upon the stability
limits, e.g. width of lateral channels.
Experiments:
Stimulus - response technique (injection of a tracer and measurement responses) has proved to be
useful for detection of cross-flow. As tracers KCl (conductivity method), KMnO4 (visualisation),
Tc99 (radioisotope) were used. Conductivity method is superior at isothermal flows, while
radioisotopes are better when the liquid is heated (KCl solution increases conductivity and
therefore electric power). Asymmetries of flow are better observed by thermometers arranged
along the lateral channels. Experiments were performed for 3 different widths of lateral channels
(18, 10, 7.5 mm), full and perforated electrodes, flow-rates 20 - 80 ml/s, heating power 0 - 6 kW.
Narrow lateral channels increase cross-flow, but only at isothermal flow
EXPERIMENTAL SET-UP
DIRECT
OHMIC HEATER
Flow
-rate
Outlet co(t)
Cross-flow is lower for wide lateral channels.
Inlet ci(t)
Impulse
response
Fig.6
Conductivity
method.measured
Experimental
set-upmethod (KCl)
•conductivy
•radioisopes (Tc 99)
Temperatures measured by
•12 Pt100 thermometers
•Fast optical probes ReFlex
by
Conclusions:
• Control volume model of natural convection can describe asymmetry and instability of parallel flow.
•Instabilities depend upon flow-rate, power, geometry and can be suppressed by narrowing lateral channels.
•Model and experiments predict that the cross-flow is suppressed (or even reversed) at non-isothermal flow.