Subsonic Inlet Duct Investigation

For a realistic estimate of ramjet subsonic duct thermal-structural conditions and construction approaches,  I looked at a generic engine/inlet combination,  sized at an arbitrary 1.00 square feet of combustor internal flow area.  Conditions inside that subsonic portion of the duct are more driven by the downstream combustor conditions than the upstream supersonic inlet characteristics.  That outcome is unlike the supersonic capture features and shockdown diffuser upstream.  Analyses here rely on standard (NACA 1135-type) compressible flow methods restricted to temperatures at which ideal gas assumptions are appropriate (under about 5000 F).

Inlet Duct Construction

I looked at arbitrary-but-realistic ramjet flight conditions of Mach 3.5 at 40 kft (on a US 1962 standard day) for “design”,  Mach 2.5 at sea level for a “low altitude minimum speed takeover”,  and Mach 5.5 at 80 kft for a “high-altitude / high-speed” point.  The best construction approach seemed to be a thin sheet metal pressure shell for the duct,  located on the outside of some thickness of magnesia insulation,  and a thin sheet metal liner shell that is perforated so as not to resist pressurization,  but does provide a smooth internal flow surface that is also impermeable to the injected fuel.  The fuel injection location is guessed as one duct ID upstream of the combustor entry,  so that there is time to achieve some vaporization.  The value for inlet duct ID d₂ is based on a dump area ratio A₂/A₄ of 0.50.

The pressure capability this inlet duct structure must resist is a max ramjet chamber pressure in the vicinity of 200 psig.  This would be experienced only in a transient max-speed terminal dive to sea level.  The combustor is not at issue here,  and its size is provided only for a practicality reference.  I used the typical strength and thermal conductivity values of stainless steel for estimating thickness and thermal behavior.  That would be k ~ 10 BTU/hr-ft-R and tensile stress allowables of ~1-2 ksi “hot”  and ~40 ksi “cold”.  I presumed the heat sink temperature maintained at the outer pressure shell was 100 F. 

This basic construction concept is depicted in Figure 1.  All figures are located at the end of this article. 

The insulation was presumed to be a fibrous magnesia,  similar to mineral wool,  but capable of withstanding higher temperatures.  Its “typical” thermal conductivity is about k ~ 0.0405 BTU/hr-ft-R.  That is actually a little higher than the conductivity of mineral wool,  but ordinary mineral wool is not rated to serve at temperatures at (or slightly exceeding) 2000 F. 

The outer pressure tube runs cool enough to be made of aluminum,  but there must be some sort of centering-connections between it and the inner tube,  which is very,  very hot.  Those standoffs or centering connections (not detailed here) must then survive hot,  so stainless steel is the better option.  In order to weld these to the outer tube,  it had to be stainless as well.  A simple SS 304L sheet metal tube will do nicely for the outer pressure tube,  with SS 316L standoff/centering devices that more-or-less resemble leaf springs.  This outer tube got sized at 18 gauge thickness (t = 0.0500 inch) to meet or exceed the pressure capability in “cool” conditions,  at the largest finished OD in the study. 

The inner tube need resist no pressure,  and may rest on the centering devices without solid attachment.  The study shows about 2200 F soakout temperature at the most demanding flight condition,  which is beyond the recommended no-scaling service limit of 1900 F for SS 309/310.  Some scaling will occur,  which roughens the inner surface,  but that may or may not actually be objectionable.  If scaling is objectionable ,  a non-ferrous superalloy must be used for this inner tube.  One of the alloys commonly used for afterburner parts would serve well.  The thickness I show for this part is the thinnest stainless sheet available,  30 gauge,  or t = 0.0125 inch.  The superalloy should be available in something comparable,  if it is needed.

Inner Surface Film Coefficient

Of the three flight conditions,  the highest film coefficient occurs at the lowest altitude,  while the largest possible driving temperature occurs at the highest speed,  which is at the highest altitude.  As it turns out,  film coefficient varies weakly with inner surface temperature,  while the heat transfer varies all the way down to zero if the surface temperature is fully equal to duct air temperature.  The net effect is that the largest heat transfer potential to be dealt with (average film coefficient multiplied by max driving temperature difference) occurs at that highest-speed condition.  A brief summary of those heating potential data is given in Figure 2 below. 

The mild film coefficient variation is shown in Figure 3 below (note scale break !!) for the high speed / high altitude condition.   Plot shapes at design and low takeover are similar,  but not shown.  The heat transferred at design is shown in Figure 4 below.  Plot shapes at design and low takeover are similar,  but these not shown here.

These were computed from a diameter-based Reynolds number ReD evaluated at bulk flow conditions (static temperature T₂ and pressure P₂,  and velocity V₂,  with density from the ideal gas equation of state).  For the other properties,  I used correlations as good for combustion gases as air,  instead of real tabulated air values.  This was as much for convenience as anything.  The Nusselt number correlation is:

                NuD = 0.027 ReD^0.8 Pr^1/3 (µ/µ_s)^0.14

for which h = NuD k / D,  with k also evaluated at bulk flow conditions.  The two viscosities sown in the equation are µ evaluated at bulk flow conditions T₂,  and µ_s evaluated at surface temperature T_S conditions.  The heat flux equation is Q/A =  h (T₂ – T_S).  Conditions are rather subsonic (well under M₂ = 0.7),  so compressibility and dissipation are simply not large-enough issues to warrant modeling. 

I used the average film coefficient h without any final-T_S correction in the subsequent cylindrical heat transfer model,  because the variation of h with T_S is so mild.  This is good enough to find out “what ballpark” we are playing in.  The cylindrical-geometry heat transfer model has fluid at bulk temperature T₂ transferring heat to the inner surface at T_S through that film coefficient .  That heat conducts through 3 concentric layers:  the inner layer is metallic and thin,  the middle layer is insulative and thicker,  and the outer layer is metallic and thin.  That outer layer’s outer surface is presumed to be held at a constant heat sink temperature T_sink,  in this case,  100 F.

This sort of 3 layer construction with insulation sandwiched between two metal layers is actually quite practical,  if the inner layer is vented so as not to hold duct pressure.  That makes it a structurally-unloaded piece of thin sheet metal,  serving only to be a smooth surface for the air flow,  and an impermeable surface for the injected fuel spray.  It being hot and weak is then irrelevant.  The cool outer layer is the actual pressure shell for the duct,  and because it is cool,  it is much stronger,  leading to a thinner,  lighter part.  The real variable to investigate is the insulation thickness:  we are trading off higher inner surface temperature for lower heat transfer to the sink at thicker insulation.  That insulation must be fibrous or at least open-cell,  so that pressure can instantaneously equalize right through it.

Layered Conduction Model

The heat transfer model is a textbook cylindrical geometry,  as shown in Figure 5 below.  It works by summing thermal resistance terms for the film coefficient and the three layers,  and is formulated to determine heat transfer rate Q per unit length of inlet duct L.  The cylindrical geometry shows up in the logarithmic variation in terms of layer radii,  and the 2 pi factor.  The individual thermal resistance terms can be used to determine the temperature drops through each layer,  including the thermal boundary layer represented by the film coefficient:

                Q/L, BTU/hr-ft  =  2 pi (T₂ – T_sink) / [denominator]

              where denominator = 1/R₂ h  + sum of {ln(ro/ri)/k} for the 3 layers

                and R₂ = 0.5 d₂;  with ro = ri + t for each layer

As shown in the figure,  the inner metal layer has ri = R₂,  with ro = ri + t for the metal.  That metal ro is the ri for the insulation layer,  with its ro equal to its ri + t.  The ro for the insulation is the ri for the outer metal layer.  Its ro is that ri plus t.  The OD for the layered inlet structure is then just twice the ro for that outer metal layer.

As a nod to the notion of heat-sinking that outer layer,  it is instructive to compute a Q for a representative duct length,  in this case L₂/d₂ = 1.00 to allow adequate length to spray and vaporize fuel. This answer is appropriate to heat-sinking into adjacent structure,  which would have to be in intimate contact over all of the duct outer surface.

Otherwise,  that outer duct surface would have to be liquid-cooled with some sort of jacket.  If one divides the heat flow by the product of liquid coolant heat capacity c and the allowed temperature rise dT,  one obtains the coolant flow rate wc.  Multiplying that by a suitable flight time gets the mass of coolant fluid required,  and by means of a density,  the volume of that coolant. 

I used c = 0.5 BTU/lbm-R as typical for a hydrocarbon fuel,  and dT = 20 F as “reasonable”.  Flight time was assumed 1000 sec as “typical”,  and liquid specific gravity is 0.8 for a typical kerosene-like hydrocarbon fuel.  These values are not-necessarily-at-all “right”,  but they are realistic enough to see informative trends in the answers.

Results

Figure 6 below shows the trends of heat rate to be dealt with (Q), coolant volume required (V),  and insulated duct OD,  all vs insulation layer thickness.  If you look at the OD trend:  at about 3 inches thick,  that finished duct size matches the combustor OD size for a 1.25 inch case-plus-ablative thickness allowance.  That sets the max feasible duct insulation thickness at 3 inches,  in a very real and practical sense.

Both Q and V decrease very rapidly with thickness from 0.5 inches to 1 inch,  then not so fast,  from 1 inch on thicker.  In a practical sense,  then,  1 inch insulation is about the thinnest insulation we should consider.  Thus,  one has the apparent design freedom to choose from about 1 to about 3 inches of insulation thickness,  in this 3-layer approach.  But,  bear in mind that heat rates and coolant requirements are actually a little larger,  due to the extra conduction paths afforded by the centering standoff structures that support the inner metal layer.

The thermal-structural design ranges are not so sensitive to insulation thickness,  as shown in Figure 7 below.  At any practical insulation thickness,  from half an inch on up,  the heat rate is reduced enough by the simple presence of insulation,  to limit the temperature drop across the thermal boundary layer to trivial values.  In effect,  to within just a few degrees,  the inner metal shell soaks out steady state to the inlet bulk air temperature T₂ = 2228 F,  which at subsonic velocities is,  in turn,  very close to the inlet total temperature T_t2 = 2803 R = 2343 F. 

Model results vary from T_S = 2188 F at half an inch,  to 2222 F at 4 inches.  In effect,  the lesson here is that one can use the inlet total temperature T_t2 as a good guide to suitable material selection for that inner shell,  and also just how hot the inner layer fibers of the insulation will get.  T_t2 is easy to compute from only flight Mach number and the outside air temperature at altitude.  Such is given in Figure 8 below.  The value of temperature for which these calculation methods fail (not being ideal gas anymore) is also shown.  One can derive speed limits from that,  outside of which predictions made by these methods will simply not be accurate. 

The speed limit for that “not air” aeroheating-analysis technique limitation is about Mach 8 in the cold stratosphere,  and about Mach 7 at sea level,  and 160 kft,  where the outside atmospheric air is about as warm as at sea level.  Good non-scaling max service temperatures would be 1200 F for SS 304/304L,  1600 F for SS 316/316L,  and 1900 F for SS309/310.  By way of comparison,  both titanium and plain carbon steel are listed as about 750 F max service.  There are several alloy steels capable of service to about 1400 F,  but only one also has very high cold strength:  17-7PH (that is what makes it suitable for ramjet cases that must also serve as integral boosters).  The nonferrous superalloys used for afterburner parts will go to ~2000-2500 F,  but also have low cold strength.

As for the inlet duct pressure capability,  this varies mildly with insulation thickness throughout the practical range of thicknesses.  It might be possible to reduce the sheet metal thickness to the next higher gauge number at lower insulation thickness nearer 1 inch,  but this has to be traded against the risk of cracking at joints during detail design.  The sharp shape changes at joints very effectively act as serious stress concentrators.  Rise factors can range from 1.5 to ~5.  These thicknesses will be larger at larger combustor sizes,  but the trend shapes will be similar.

Final Comments

First,  these results vary with combustor size.  The data shown here are for a generic combustor of 1 square foot flow cross section.  At larger sizes,  the sheet metal thicknesses for the inlet duct will need to be thicker,  particularly the outer pressure shell layer.  But the basic behavior trends will be the same.

Second,  for the Mach 5.5 at 80 kft flight condition analyzed here,  the inlet air temperature is really too hot at ~2200 F for repeated use of SS 309 or SS 310 construction as the inner tube.  Such parts would survive,  but would experience an ever-increasing surface oxidation scaling effect that roughens the inner surface.  One of the afterburner-part superalloys would be necessary for repeated operation.

Third,  we have NOT addressed here the rest of the supersonic inlet structures.  These include the compression spike or ramp structures,  the cowl lip structures that are heated on both sides,  the supersonic throat structures,  and the supersonic-to-subsonic shockdown divergence channel that connects to the subsonic duct analyzed here.  All of these will be more demanding problems to solve than this subsonic duct.

Fourth,  we have NOT considered here the use of smooth-surfaced non-porous ceramics as the inner tube sleeve for subsonic duct construction.  Such could be used to higher flight speeds than even the nonferrous afterburner-part superalloys.  However,  these would be brittle and shock-sensitive,  and of substantially-larger wall thicknesses.  Being dense,  they are highly thermally-conductive,  thus tending toward isothermal behavior.  Once hot enough,  there is no practical way to hang onto such a hot part!

Fifth,  I did a quick check of radiation cooling capability toward 70 F surrounding structures for a duct OD sink temperature of 100 F.  The results showed Qrad ~ 0.02 to 0.03 BTU/sec when otherwise Q ~ 1.2 to 0.2 BTU/sec.  At factor 6 to 60 too small a radiative heat flow,  there just isn’t any practical help there,  at nice,  low outer duct shell temperatures.  The outer tube would have to run significantly hotter (somewhere in the neighborhood of 500-600 F),  to reach “steady state” cooling to the structure that way,  at 2 or 3 inches of insulation.  And that adjacent structure would warm rapidly,  reducing its effectiveness as a radiation heat sink.  It’s a possibility,  but probably not that practical.

Related Articles by GWJ on http://exrocketman.blogspot.com:

“A Look at Nosetips (or Leading Edges)”,  1-6-19

“Thermal Protection Trends for High-Speed Atmospheric Flight”,  1-2-19

To navigate on that site,  look for the by-date-and-title navigation tool on the left of the web page.  Click on year,  then month,  then title (if more than one article was posted that month).  If you click on a figure,  you can see all the figures enlarged.  You “x-out” to return to the article itself. 

Figure 1 – Construction Approach Concept and Principal Study Dimensions

Figure 2 – Comparison Among the Flight Conditions

Figure 3 – Mild Variation of Film Coefficient vs Inner Surface Temperature (Scale Break!!)

Figure 4 – Strong Variation of Heat Transferred with Inner Surface Temperature

Figure 5 – Cylindrical Geometry Thermal Conduction Model

Figure 6 – Variation of Heating and Cooling Parameters with Insulation Thickness

Figure 7 -- Variation of Thermal-Structural Parameters with Insulation Thickness

Figure 8 – Inlet Air Total Temperatures vs Speed and Altitude,  for Selecting Inner Shell Materials