Sunday, May 20, 2012

Recommended Broad Design Guidelines For Valveless Pulsejet Combustors

Update 6-5-2016:  this has to be one of the most,  if not the most,  popular article on the entire website.  Clearly there is a community out there that experiments with these things.  I hope what I posted was useful to you.
Recommended Broad Design Guidelines For Valveless Pulsejet Combustors  
GWJ       3-27-12     rev.1   4-21-12    rev.2    5-20-12
rev. 3 5-27-12 puts the appendices into a readable form

Update 1-19-13:  This stuff was just too much fun to play with,  when I helped Justin Friend with his engines.  I must build some of these things myself.  That will require a sheet metal capability not currently in place in my shop out here on the Idea Farm.  But as soon as I do some experiments,  I will report them here on "exrocketman".  Just don't hold your breath!

Related articles also posted here include:
4-30-12      Big Student Pulsejet an Even Larger Hit at TSTC
                   big engine at TSTC
3-6-12        Student Pulsejet a Hit at EAA Meeting
                   small engine on golf cart
11-12-11     Student Pulsejet Project
                    small engine at TSTC

(all figures and tables at end of article)         

Designs for valveless pulsejets of several types have appeared on the internet as hobby devices.  These are generally based on designs that began appearing about 1909,  and which culminated about 1964,  as potential industrial or military items.  Some reports are now available on the internet,  that describe the many geometries tested long ago.  The premier examples are the folded geometries developed by Raymond Lockwood at Hiller Aircraft (ref. 1, 2),  based on French work,  and since publicized by his son.  

In this document,  the scaling proportions of the three Lockwood combustor designs are correlated versus combustion chamber size,  using the published Lockwood data.  Scaling parameters were suggested by Engebretson’s thesis (ref. 3),  in which several designs by J.W. Belter were successfully correlated for resonant proportions at one chamber size.  Chamber size as the independent variable is suggested by the fact that everything but chemistry can be scaled in some way with size.  Lockwood’s smallest test articles could only be run on acetylene-oxygen,  not propane-air,  which proves that point.  

Design details like flare geometries,  fuel injection nozzles and locations,  and starting air introduction techniques,  were not correlated.  The fundamental objective here was only a set of basic combustor tube geometries,  which might be expected to resonate well.


This document concerns itself with only the core combustor.  No ducting or augmentor devices are considered.  The Lockwood front entry single inlets are the basis of the correlation.  The relevant parameters so developed are applied to design classes resembling the side-entry Logan forms,  the multi-inlet/rear-entry Thermo-Jet forms,  and even annular reversed inlet designs.  

Not all parameters involved in the design are covered by available data,  or these correlations.  In particular,  Lockwood’s work indicated that some sort of flare on both the inlet and the exhaust helped resonance,  starting,  and performance.   However,  he gave insufficient data to correlate anything.  

Nothing in Lockwood’s documents suggests a scalable guide to how to implement those flares,  excepting a radius ratio of 22% for his HC-1 unit.  There is one comment about a 1/8-inch radius exit flare on one unit,  but no indication that other flares were attempted,  or that any flare geometries were scaled up or down in size.  

The only guidance we have in Lockwood’s documents for combustion chamber (only) length to diameter ratio is a relative constancy in the scaled proportions for the HS series.  His tailpipe taper rates appear to have been held constant as he scaled down.  

That limits correlatable scaled proportions to those describing the overall engine slenderness ratio,  the minimum tailpipe diameter ratio,  the inlet length ratio,  and the inlet to chamber size ratio (here correlated as area ratio to cover multiple or annular inlet designs).   Taper rates and chamber-only length to diameter ratios are simply recommended as constants.  

Lockwood Designs

These come from a Navy-funded program about 1959 that developed the “Pulse Reactor” (ref. 1).  That device was a combination of a very highly-rated valveless pulsejet combustor,  folded into a U-bend shape,  with venturi-type augmentor tubes on the inlet and outlet.  These devices were all about 9.1-inch chamber inside diameter in size.  With the augmentors,  maximum thrusts near 500 pounds were demonstrated,  along with minimum TSFC’s of about 1 pound per hour fuel flow per pound of thrust.  Without the augmentors,  thrusts were about factor 2.2 smaller,  and TSFC’s about factor 2.2 larger.  

Lockwood’s original “Pulse Reactor” combustor was a variation on a French design developed at SNECMA.  Hiller Aircraft and SNECMA were teamed up on this old Navy program,  and Lockwood termed his version of the basic French combustor the HS series.  The French called this folded design “Ecrevisse”,  which translates as “Crayfish”.  

In the early 1960’s,  Lockwood conducted tests of scaled-down combustors,  starting with scaled versions of the HS series.  This was an Army program,  culminating in a report published in 1964 (ref. 2).  During that effort,  Lockwood investigated two other variations:  a conical combustor geometry he designated the HC series,  and  a cylindrical combustor with a much sharper transition to the tailpipe,  that he designated the HH series.    See Figure 1.

These combustors all feature an inlet larger than the minimum size of the exhaust passage.  This is because he needed no “aerodynamic valve” effect to limit backflow out the inlet during the pressure (“spit”) phase of the cycle.  Lockwood folded his designs 180 degrees,  so that both inlet and exhaust contributed directly to total thrust.  That fold is not depicted in the figures.  This larger inlet apparently allowed the combustor to ingest far more air than the designs of others,  apparently leading to Lockwood’s devices being so very highly-rated for thrust and fuel consumption.  

It appears in the Navy report that Lockwood ran most of his 9.1 inch tests on propane.  In the Army report,  Lockwood reports that most of his scale-down work also baselined propane fuels.  There were attempts in the larger sizes to run on gasoline.  The smallest combustors would not resonate on propane-air,  but could be made to run on separate injection of oxygen and acetylene.   Lockwood gave few details on fuel injection nozzle design and location,  and essentially none on feed pressures.  

Information on the minor geometry changes to “tune-up” the resonant shell geometry was almost completely lacking.  The exceptions were one lip radius to inlet radius ratio on combustor HC-1,  the notation that a small-radius exit flare helped strengthen resonance,  and that an inlet tapered 2 degrees to converge in the efflux direction helped starting and allowed reductions in tailpipe length (the details of which he did not document in that report).  

These lacks are why the design correlations recommended here are for the gross overall combustor tube dimensions.  Details of flare radii,  fuel injection,  and starting air are not addressed in this document.  

Engebretson’s Thesis

In 1965,  Roger Engebretson published a master’s thesis at North Dakota State University on an investigation of a common class of valveless pulsejets (ref. 3).  These were developed from a series of successfully-pulsating designs by J.W. Belter.  Engebretson’s project was aimed at investigating possible propulsive application of the Belter combustors,  originally developed to be air heaters.  

Engebretson did not scale up or down in size.  He did look at 3 possible inlet tube locations,  and two inlet tube sizes.  One of the side-entry inlet locations did not work at all,  the other did,  with little performance difference between it,  and the front entry location (which was similar to Lockwood).

The main difference between Engebretson and Lockwood is that Engebretson’s inlets were rather small in proportion to the combustor diameter,  unlike the Lockwood designs.  Engebretson did this to retain the “aerodynamic valve” action of an easier “spit” out the larger exhaust passage.  As a result,  his thrust levels remained a lot smaller for the same sizes than Lockwood’s.  

Engebretson stated his TSFC’s were much higher than Lockwood’s,  but a close reading indicates he was comparing his plain-tube TSFC to Lockwood’s augmented “Pulse Reactor” devices.   Engebretson’s TSFC data are not far from those of Lockwood,  if one takes care to compare plain combustors.   Only his thrust per unit size was a lot smaller,  which this author takes as evidence of less air mass ingested through the smaller inlet,  during the intake phase of each cycle.  

In this document,  Engebretson’s data are plotted on my scaling correlations for the Lockwood data.  The sizing variables I chose were inspired by what Engebretson investigated.  I made these correlations versus combustion chamber diameter,  to reflect the fact that chemistry does not scale.  

The Engebretson combustors were simple cylindrical chambers,  with a moderate taper to a smaller straight exhaust pipe.  Engebretson never scaled up or down in size,  and he never changed the proportions of chamber versus tailpipe diameter.  See Figure 2.

Engebretson gave a little more data than Lockwood regarding the placement of fuel,  starting air,  and spark plug.  He used both gasoline and propane fuels successfully at his 4-inch size.  Further,  he notes that these same basic designs burned a mix of propane and coal dust for Belter.  Engebretson used both inlet and exhaust flares,  but gave no information about them.  

Data Correlations

For my correlation,  I pulled data from Ref. 1 for Lockwood’s 9.1-inch diameter HS-1 tube,  and the four scaled-down configurations that Lockwood reported as “actually successful” in Ref. 2.  I put these into a Microsoft “Excel” spreadsheet (Appendix 1 attached),  and processed those data into my scaling parameters.  Some of the scaling parameters are plotted in Figure 3 below.  These include the overall engine slenderness ratio Leng/Dch,  the tailpipe contraction ratio as both Dmin/Dch and Dch/Dmin,  and the inlet area ratio Ai/Ach.  These dimensional parameters are defined in Figures 1 and 2.  

It is apparent from Figure 3 that the slenderness ratio “falls off a cliff” below 3 inch chamber diameterThis matches quite well with Lockwood’s descriptions of the difficulties he had making his smallest combustors burn at all.  This is what I believe to be the chemistry effect:  reaction rates do not scale,  while cycling rate increases.   Further,  it is easier to see the trend of tailpipe contraction expressed as Dch/Dmin than Dmin/Dch.  The scale is wrong to see the inlet area ratio trend properly.  So,  that is given in Figure 4.  Likewise,  the inlet length ratio is plotted with an appropriate scale in Figure 5.  
I have included on these figures the relevant Engebretson data.  Remember,  his engines were not as highly-rated,  being limited in inlet size,  and presumably limited in ingestible airflow during the “suction” phase.  

Note that in both Figures 4 and 5,  it is easy to see for all the variables the same “fall off a cliff” behavior at 3 inch diameter,  as was apparent in Figure 3 only for the slenderness ratio.  All these figures point to the same problem below a combustor diameter of 3 inches.  This is (in my opinion) chemistry and mixing unable to complete at the faster cycle times of the smaller (shorter) combustors.  

Note that Engebretson’s resonant length correlates pretty well with Lockwood’s,  as slenderness ratio in Figure 3.  We would actually expect that outcome,  whether the combustor was “highly-rated” or not,  because this exhaust length is the quarter-wave resonant length driving the pulsation cycle.  

The inlet is non-resonant,  and is “driven” by the combustor cycle.  It necessarily must be fairly short so that its inflow purges the combustion chamber and recharges it with fresh air.  Too short,  and the inlet and exhaust inflows meet too far “downstream”,  too long and these flows meet too far “upstream”.   In my opinion,  the Engebretson thesis misjudged this to be a “resonant effect”.  At about 0.13 in the figure,  the non-integer 7.5:1 length ratio is not a resonant ratio as we would normally think of it.  

What one wants is for these colliding flows to “meet in the middle” of the combustion chamber.  The fuel, air,  and the hot residual exhaust gases all mix,  thus providing ignition for the next combustion cycle.  Fuel injection style and location,  and fuel type,  critically impact this ignition process.  Many reports indicate that this can “make or break” resonant operation entirely,  no matter the geometry. 

The best “eyeball” curve fits obtained from reading these graphs are summarized in Figure 6 below.  Note the redundant equations for tailpipe contraction.  Because of the larger slope constant,  I prefer the Dch/Dmin form.   These were all formulated by reading the graph at 3 and 9 inch diameters,  assuming the evident linearity.  The reference location was 3 inches in each equation.  The basic form is the point-slope form of the linear equation in two variables.  Nothing is valid below 3 inches chamber diameter.  

Discussion of the Correlating Parameters

As defined in the figures above,  my length and diameter definitions may differ somewhat from those in the three references.  I tried to pick definitions that made sense in terms of the physical processes I know to be happening.  

In particular,  I picked the tubular length of the inlet tube as inlet length Li,  exclusive of the axial length of any tapered bulkhead into the combustion chamber.  This is because during intake,  the sharp angle at the inlet/bulkhead joint will induce flow separation,  taking one from a transient undeveloped pipe flow,  into a chamber filled with recirculation dead zones and residual gases.  

I did include the axial length of any inlet flare in the inlet length.  This is because the intake phase uses this flare to smooth out the flow separation disturbances that a sharp tube entry would induce.  It simply is part of the length covered by the entering flow.  There is flow separation at the flare joint on the efflux phase:  that flow does not follow the curve of the flare,  at least not very much.  

The rest of the overall (straight or folded-centerline) length is the engine (resonant) length Leng.  The forward chamber bulkhead is the pressure node for the quarter-wave resonance.  That length includes the chamber,  the axial lengths of any conical bulkheads and any tapered sections,  the entire exhaust tube,  tapered or not,  and the axial lengths of any exit flares.   I include the exit flare length in the engine length for the same fluid dynamical reasons as the inlet flare gets included in the inlet length.

Chamber diameter Dch itself is an inside diameter (of little effect in thin sheet metal construction),  taken to be the largest diameter,  if the chamber is tapered in any way.  

There is in any of these designs at least a short length of minimal-diameter straight tube in the exhaust pipe.  This is Dmin,  and it is an inside diameter.  It and the inlet size represent the two impedances to efflux massflow during the exhaust phase.  

The inlet diameter Di associated with inlet area Ai is also an inside diameter,  and taken to be the larger diameter at chamber entry,  if there is taper.  This is more for construction dimensional control than fluid dynamics.  If the inlet tube really is tapered,  that taper is supposed to be in the efflux direction (according to Lockwood),  making the possibly-flared end of the tapered tube a little smaller.  However,  at 2 degrees taper and a relatively short length,  there is little difference in inlet diameters at either end,  even though the physically-smaller diameter is probably what controls the fluid dynamics.  

I correlated inlet size on area instead of diameter,  to allow for the possibility of a rearward-facing annular inlet geometry to replace the 180-degree fold of the Lockwood designs.  

I did not initially correlate combustion chamber length or volume.  That is a serious lack.  I do note that the combustion chamber cylindrical and taper length proportions of the Engebretson/Belter design and the Lockwood HS-series are just about the same,  and each of these is about 2 to 2.5 times the chamber diameter.  

How much of this total length is actual “combustion chamber” volume,  and how much is just a gentle entry to the exhaust tube,  is purely a matter of speculation.   The Lockwood HH-series has only a chamber length without a gentle taper,  of the same general dimensions as the cylindrical chamber of his HS series.  Lockwood’s HC-series has a straight conical chamber of roughly the same length as the combination of chamber and gentle taper in his HS series.  

Recommended Designs

 Design recommendations were formulated for three geometry classes,  to be based on these correlation equations.  The independent variable is combustion chamber inside diameter,  to be 3 inches or greater for propane.  Be very careful extrapolating beyond 9.1 inches diameter,  the results look increasingly unrealistic beyond about 10 or 12 inches.  The numerical data are appended below as Appendix 2.  

One of the three geometry classes are front-entry inlets similar to the Lockwood designs.  How to do that is illustrated in Figure 7 below,  for designs resembling Lockwood’s HS,  HH,  and HC series.  This figure shows the taper rates,  what we do know about flares,  and the best approximations I could determine for chamber lengths (as ratios to Dch).  

The second basic geometry class is repositioning the front inlet (Lockwood type) to side inlet (Logan type) and aft inlet (Thermo-Jet type),  as shown in Figure 8 below.  The basic combustor and tailpipe is unchanged.  For side entry,  a more aft location (about 75% of cylindrical chamber length) is the one that worked for EngebretsonFor aft entry,  the inlet needs to be divided into 2 or more separate inlet tubes,  spaced circumferentially,  and preserving the total inlet area ratio.  Taper and flare do apply to each inlet separately.  

The third basic geometry is an annular aft-facing inlet,  made of a shell surrounding the basic combustor,  which has a blind end on the front.  This is shown in Figure 9 below.  The idea here is better packaging for a flight engine.  Air uptake is in the annular space,  and makes a 180-degree turn as it enters the combustor at the front end.  Inlet area ratio as correlated must be preserved in the annulus and at the front end.  The inlet length is essentially from front centerline to flare lip,  as shown,  plus a radius inward,  and must match the correlated inlet length ratio.  That last inlet length recommendation is actually very speculative,  and needs testing verification! 

What Is Not Covered

These correlations and recommendations cover the shape and proportions of valveless pulsejet tube geometries that are likely to be robustly resonant as first built,  provided that appropriate fuel injection is installed.  My best interpretation of Lockwood,  Engebretson,  and some others,  is that inappropriate fuel injection can cause an otherwise acceptable geometry to operate poorly,  or even not at all.  Further,  liquid fuels behave completely differently from gaseous fuels.  Fuel injection is necessarily a large separate topic.

The details of flare shape and sizing are poorly covered in these reports.  I could correlate almost nothing for actual recommendations.  This area needs experimentation to define the effects.  It is known that some sort of flare is beneficial,  even critical,  to tube operation.  That means the acoustical wave trap afforded by the flare on a musical instrument is having a beneficial effect on the resonant properties of the pulsejet tube.  A crude approximation (from a very old issue of Scientific American magazine)to the wave trap impedance (the “horn function”) is: 

                F(x)  =  1/[R*r]   where r is the flare radius and R is the tube radius at that same axial x

The details of starting air injection are better covered by Engebretson than Lockwood in these old reports (Ref. 1, 2,  and 3).  Even so,  there is almost no information available beyond what I put in Figure 2 above.  The feed was ordinary “shop air” at something like 85 to 100 psig.  I make no recommendations,  beyond the use of a leaf blower into the inlet,  based on what I see on the internet.  

The details of spark plug location are not well covered in any of these reports.  An educated guess says that a continuous spark source might have advantages over an intermittent device,  based on the interaction potential of the spark pulse frequency with the engine combustion cycle frequency.   No recommendations are made for that topic here.  

Some old sources point to the criticality of the timing of the fuel-air explosion relative to the “suction” phase inflow phenomena.  At full thrust in a highly-rated engine,  this suction is about half an atmosphere below ambient.  That is what drives the inflows.  

These inflows up the inlet and exhaust cause a fluid collision in the combustion chamber.  That fluid kinetic energy is converted to a small pressure rise,  thus there is a slight compression before ignition.  The explosion,  acting against the inlet and exhaust fluid columns,  is thus “confined”,  so that it produces a much higher chamber pressure.  That final pressure is about 2 atmospheres above ambient at maximum thrust in a highly-rated device.  It drives the “spit” flows out both inlet and exhaust.  

If the fuel-air explosion occurs too soon,  or too late,  pre-compression is lost,  and performance drops dramatically.  It is even possible the pulsejet will fail to operate at all.  Fuel injection velocity,  direction,  location,  and feed pressure all affect this ignition “timing”.  So does fuel type,  since there is both a physical delay,  and a demand for considerable heat,  to achieve vaporization with liquid fuels.

In his 1930-vintage“Pulse Pot” combustors,  Francois Reynst addressed this “timing” issue by means of clever fluid dynamical control.  Reynst used a guide venturi inside the combustor to force vortex formation on the blind end of his combustor.  As long as fresh air inflow persisted,  the vortex spun and grew,  but seemed not to ingest enough of the surrounding hot gases and fuel vapor to ignite.  Once inflow ceased,  the vortex continued spinning on momentum,  but the fluid flow “breakaway” (as fresh air inflow stopped) caused it to suddenly ingest large amounts of the surrounding hot gas plus fuel vapor,  leading to a sudden ignition and explosion of that vortex.  See Ref. 4.

Position of the spark source can also affect the explosion timing,  once pulsation starts.  An inappropriate spark location can cause late or early ignition” timing”,  according to some sources.  These effects can preclude operation,  or at least severely degrade performance.  

Again,  I make no recommendations regarding fuel injection details,  spark timing and location details,  or starting air details in this document.  

Addendum for Revision 1 – Thrust Intensity Scaling

To the data in Appendix 1,  I recently added some thrust scaling data.   My best estimate of actual “combustion chamber volume” would be about two chamber diameters long,  based on the Lockwood HH-series dimensions,  and on the Engebretson/Belter design.  

Using Lcomb = 2*Dch,  and thus Vcomb = pi*Dch3/2,  I computed some thrust per unit volume data vs chamber diameter.  Thrust was total for the inlet and exhaust acting together,  but with no augmentors.  These data are given in Figure 10 below.  They show little or no correlation.  

Then,  because thrust and peak cycle pressure are said to be related,  I tried thrust per unit chamber cross sectional area,  computed as pi*Dch2/4.   These correlated very well,  as given in Figure 11 below.  The correlating equation “eyeballed” from these data,  for only diameters 3 inches and up,  is 

                Ftotal/Ach lb/ = (D – 3 inches)*0.35 + 1.6

Again,  this “falls off a cliff” below a chamber diameter of 3 inches,  reflecting what this author believes is insufficient reactivity of propane fuel at the fast cycle times of the higher-frequency miniature sizes. 
A similar affliction should occur with volatile liquid fuels like gasoline,  except starting at a larger diameter.  This corresponds to a longer engine length,  lower frequency,  and longer cycling time.  

Thrust Augmentation

Based on Lockwood’s work,  thrust multiplications of 1.4 to 2.2 should be possible at otherwise the same geometries and fuel flows,  given properly-designed and properly-positioned venturi-type augmentors on both inlets and exhausts.  Thrust in this sense is the sum of both inlet and exhaust,  implying a folded or reverse-inlet engine geometry.  The details of exactly how to design the augmentors is beyond scope herein.  

  1. Lockwood,  Sargent,  and Beckett,  “Thrust Augmented Intermittent Jet Lift Propulsion System”,  Hiller ARD-256,  February,  1960.
  2. Lockwood,  Patterson,  Beckett,  and Graber, Summary Report on Investigation of Miniature Valveless Pulsejets”, Hiller ARD-307 / Army TRECOM TR-64-20, February 1964.
  3. Engebretson,  “Investigation of the Performance Characteristics of the Valveless Pulsejet”, MS Thesis,  N. Dakota State University,  February 1965.
  4. Thring (ed.), “Pulsating Combustion, the Collected Works of F. H. Reynst”,  Pergamon Press,  1961.
Addendum for Revision 2 – Experimental Data (TSTC student Justin Friend)

Recent experiential data with an HS-type Lockwood combustor indicates the definite positive influence of significant inlet and exit flares of a “horn function” type of shape.  These were hose-clamped articles of slit tube bent to a constant “large” radius (more than 25% of local tube radius).  The slit gaps were filled with bent triangles of the same sheet metal,  welded in place.  

Fuel injection also proved quite critical,  even with propane gas fuel.  The final injection rig that worked was designed to stop flow at about 2 atm local pressure.  This tube included both carefully-sized outlet holes,  and a surge chamber Tee’d into the line,  fed directly from an ambient propane tank.  No reliable final dimensional data are currently available.  The near-10 inch chamber diameter HS-pattern engine failed to throttle up until the fuel injection tube matched this pattern,  however.  

This propane injection geometry was 3/8 inch OD copper tubing inserted axially into the inlet,  and clamped to one side.  It featured 4 injection holes in 2 opposed-hole pairs,  spaced about 2 inches apart axially.  These holes were in the neighborhood of 1/8 inch diameter,  which is a sensitive variable.   These holes were arranged to inject past the start of the 45 degree inlet bulkhead,  another sensitive variable.  

This large HS-pattern engine was sized with a correlation that this author did over 30 years ago.  That correlation is not the same tube geometry correlation as is recommended herein.  In particular,  the exhaust-side minimum tube diameter looks to be too large.  This difference may explain the experientially-noticed lower-than-expected thrust.   No hard measurements are available,  however.  

Figure 1 – Basic Parameters of the Three Lockwood Valveless Pulse Combustor Designs
 Figure 2 – Basic Parameters of the Belter Designs Investigated by Engebretson
 Figure 3 – Correlation Plot Usable for Slenderness and Tailpipe Contraction Ratios
 Figure 4 – Correlation Plot Usable for Inlet Area (and Tailpipe Contraction) Ratios
 Figure 5 – Correlation Plot Usable for Inlet Length Ratio
Figure 6 – Curve Fit Equations Derived from the Data Correlations
 Figure 7 – Applying the Correlations to Front-Entry (Lockwood) Designs
 Figure 8 – Extending the Correlations from Front-Entry to Side-Entry and Aft-Entry Designs
 Figure 9 – Extending the Correlations to Annular Aft-Entry Designs
 Figure 10 – Non-Correlation of Total Thrust Per Unit Combustor Volume
Figure 11 – Good Correlation of Total Thrust Per Unit Chamber Area

Appendix 1 Correlation Data - Part 1 of 3

Appendix 1 Correlation Data - Part 2 of 3

Appendix 1 Correlation Data - Part 3 of 3

Appendix 2 Estimates vs Size Based on Correlation Equations


  1. Nice Blog ! Thank you for your very nice articles. I look forward to visiting your site in the future!
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  2. Thank you for your kind comment, Ms. Borua. That (useful articles) is exactly what I try so hard to achieve.

    If you visit this particular article again, please not that I upgraded the appendix tables into something folks could read.

    I had to retype the data into another piece of software in order to manage this correctly. Tables input into Word from Excel do not transfer correctly on Blogspot.