4. Physical Problem of PCS-2000 Setup and Solution Methods
4.2 The Problem of Measuring Correct Ice Particle Sizes Using the Sampling Geometry of PCS-
4.2.1 The True Gas Temperature in the Sampling Tube of PCS-2000
Physical Problem of PCS-2000 Setup and Solution Methods
4.2 The Problem of Measuring Correct Ice Particle Sizes Using the
Physical Problem of PCS-2000 Setup and Solution Methods
In the first model, the heat transfer to the gas starts as soon as the gas enters the tube. Both the velocity and the thermal boundary layers begin to develop immediately and simultaneously. These regions are characterised by the hydrodynamic entrance length Lh and the thermal entrance length Lt which in this case are both measured from the tube inlet. This is shown in Figure A.1 in Appendix A.
In the second model, the heat transfer to the gas starts with a delay after an isothermal section.
For such a case, the hydrodynamic entrance length Lh is measured from the point where the gas enters the tube, but Lt is measured from the location where the heat transfer starts, because the thermal boundary layer begins to develop in the heat transfer section. This is shown in Figure A.2 in Appendix A.
By comparing the above two models with our problem, it is obvious that one has to choose the second model to solve our heat flux problem for thermally developing, but hydrodynamically developed conditions. The reason for this choice can be seen in Figure A.2 Appendix A: the temperature difference between the gas temperature inside the tube and the tube walls is zero or negligible in the first 585 mm of the connecting tube, sufficiently long to develop a parabolic velocity profile. The solution of this problem in the case of laminar forced convection inside a circular tube with a uniform wall surface was given by Greatz and later quite independently by Nusselt (Özişik, 1985; Welty et al., 1984).
An IDL code was written, based on the mathemathical model which is explained in Appendix A, to calculate the distribution of the gas temperature in the connecting tube between the AIDA chamber and the PCS-2000 according to the following equation
4
w g
exp
mw out T m p
T T l h
T T D u
ρ
C⎛ ⎞
− = ⎜⎜ ⎟⎟
− ⎝ ⎠ (4.1)
where Tg is the gas inlet temperature, Tw the tube wall temperature, l is the tube length and Tout is the outlet temperature at a distance l from the entrance, hm the heat transfer coefficient, DT the tube diameter, um the mean gas velocity, ρ the gas density, and Cp the specific heat of the gas at constant pressure (Özişik, 1985; Welty et al., 1984 ).
The time spent by ice particles in the connecting tube between the ADIA chamber and PCS-2000 is only a fraction of a second, ~ 0.4 s, before they are optically detected in the measurement volume of the PCS-2000. This time is short in comparison with the time until the next measurement of state variables in the AIDA chamber occurs. The length of the connecting tube between AIDA and
Physical Problem of PCS-2000 Setup and Solution Methods
PCS-2000 is divided vertically into increments of 1 mm. The following equation (4.2) was used to calculate the distribution of the gas temperature along the tube axis in time steps of one second. The gas temperature every 1 mm along the tube axis was calculated from the following equation,
Tout
[ ]
i+1
= Tw +(
Tw −T ig[ ] ) e
−⎜⎛⎜⎝D uT4ml hρmCp⎞⎟⎟⎠ (4.2) where Tout[i +1] is the exit gas temperature at the position of i+1 as a function of the inlet gas temperature at the position i while the parameters hm, um and ρ are calculated as explained in Appendix A. Figure 4.3 shows the calculated profile of the gas temperature along the tube center line as a function of time for experiment N° 74 IN02, cf. Figure A.2 in Appendix A. In the three dimensional plot the x-axis represents the running time of the experiment, the y-axis represents the distance from the tube entrance in mm while the z-axis represents the gas temperature along the center line of the tube in K.Figure 4.3: Profile of the gas temperature along the tube center in a three dimensional plot where the x, y and z axes represent the run time of the experiment, the distance from the tube entrance, and the gas temperature, respectively. Activation N°74 IN02.
Physical Problem of PCS-2000 Setup and Solution Methods
The time dependence of the gas temperature at 0 mm (tube entrance) during the running time of the experiment is of course the same as the mean gas temperature versus time, which is shown in Figure 2.13 in section 2.7 chapter 2. The temperature along the tube center line stays constant until, downstream of the hydrostatic length, it increases slowly with the beginnig of the heat transfer section (cf. Figure A.2 in Appendix A), and much more strongly with the beginning of the third thermal section.
The accuracy of this calculation was examined by plotting in Figure 4.4 the difference between the calculated gas temperature at 850 mm from the tuber entrance and it is value measured by the thermocouple at the same position. Figure 4.4 shows that the model underestimates the gas temperature at this position by a few degrees. It should, however, be noted that the position of the thermocouple is not exactly at the tube center, and therefore does not measure precisely the calculated gas temperature which applies to the tube center. The oscillations of Tdown in Figure 4.2d, which are also superimposed on ∆T = Tdown,meas – Tdown,cal in Figure 4.4, are caused by the heat source in the box which encloses the PCS-2000 instrument: it is switched on and off at regular intervals by the temperature controller. These oscillations are neglected in the model.
0 500 1000 1500 2000
Time (S)
−5 0 5
Tdown_meas−Tdown_cal (K)
Figure 4.4: Difference ∆T = Tdown,meas – Tdown,calc between the measured outlet temperature and the calculated gas temperature at a distance of 850 mm from the sampling tube entrance.
Activation N° 74 IN02. The temperature fluctuations are due to the heat source in the PSC-2000 box which is switched on and off by the temperature controller.
Physical Problem of PCS-2000 Setup and Solution Methods