Development of Coupled Flowfield - Radiation

Solution Methods in Ablative Environments

Capt Jeffrey R. Komives∗ and Dr. Robert B. Greendyke†

Air Force Institute of Technology, Wright-Patterson AFB, Ohio, 45433, USA

A seven-species hypersonic flow solver that independently tracks vibrational energiesof diatomic species has been coupled with SPRADIAN for the purposes of developing aspectrally accurate coupled radiation / flowfield solution. Results are presented for theFIRE-II and RAM-C geometries, and compared to experiment results where available.Shortfalls of the method are identified, along with avenues of further research. It is desiredto eventually develop the capability to use a complete air chemistry set with ablationbyproducts for a total assessment of the radiation environment around an arbitrary shape.

Nomenclature

Cf = activation energyD = effective diffusion coefficientE = approximate electric fieldN = number densityP = pressureQ = energy exchange between modesT = temperatureV = effective diffusion velocityZ = ionic valencyω̇s = species generation or destructionη′ = thermal conductivity coefficientτ ij = viscous shear stressθd = characteristic temperature of dissociationε = elementary electronic chargee = specific energykf = forward rate coefficientq = heat fluxu = velocityx = position component

αdir = overlap in radians between transmission direction and cell neighborβ = total included angle of transmission directionε = coefficient of emissivityκ = coefficient of absorptivityρ = densityc = view coefficient from cell to given neighbor in direction dirI = radiative intensityM = Mach numbers = radiation propagation distance

∗Masters Student, Department of Aeronautics and Astronautics, AFIT/ENY, 2950 Hobson Way, Student Member AIAA†Associate Professor, Department of Aeronautics and Astronautics, AFIT/ENY, 2950 Hobson Way, Associate Fellow AIAAThe views expressed in this article are those of the author and do not reflect the official policy or position of the United

States Air Force, Department of Defense, or the U.S. Government.

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AIAA 2009-1031

This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.

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Subscripts∞ = free streamλ = function of wavelengthi, j = cell indicesn = neighboring cells = speciesv = vibrationw = walleff = effectivebb = black-body

Superscriptsi,j = ith and jth components in orthogonal coordinates

I. Introduction

Vehicle flight at hypersonic Mach numbers results in substantial heating to the forebody of the flightvehicle through convective and radiative heat flux transfer mechanisms. The radiative transfer of energy

can represent a substantial portion of the overall flux to the surface of such a vehicle, accounting for as muchas 60% or more of the total heat load in a blunt body. This effect is not limited to blunt bodies however, butcan also affect the stagnation region of narrower bodies, causing substantial heating and/or ablation to thenose tips of the thin vehicles. Due to the nature of hypersonic flow, even thin leading edges will become bluntat some point in the vehicle trajectory, thereby leading to increased radiation effects. Unfortunately, thesolution of radiative heat transfer is a very complex area of study that is poorly understood and rarely utilizedin the flowfield solutions of most computational codes that are used to solve the flowfields of such vehicles.The effect of radiation is of particular concern for the United States Air Force with increased emphasis uponthe ability to have prompt, reusable space vehicles. Radiation is always a concern in hypersonic vehicles,but the need to have reusable vehicles may require the use of ablative and replaceable thermal protectionsystems composed of ablative carbon materials, which presents interesting challenges for designers. Carbonis both a high emitter of radiation as well as a high absorber of radiation, therefore making the net effecthighly variable depending on the vehicle shape, velocity, and altitude. It is possible that carbon materialsablated at the nose of a vehicle could transport their energy downstream in the form of internal energymodes and re-radiate the energy at locations downstream even over narrow bodies that typically do nothave significant radiative emission due to their shape. No “rule of thumb” can discard this possibility in anablative environment.

Radiation, as a flowfield phenomenon, is complex for many reasons. One major difficulty is that radiationpropagates at the speed of light, or virtually instantaneously in the computational domain of the averageflight vehicle. The speed of propagation can be used to an advantage in computational simulation in thatthe radiative calculation can now be separated from the flowfield solution in a temporal sense. Radiationalso propagates by line-of-sight, not flow direction, raising geometric difficulties for the code developer inaccommodating radiative effects into the conventional grid-based flowfield solution methods since line ofsight for each computational cell has to be determined. The thermophysical processes that lead to radiationalso present a unique set of challenges. Radiation is a function of (T 4) - making accurate temperaturedetermination a necessity in the flowfield. It is also very sensitive to which temperature is being consideredif in a thermal nonequilibrium environment - the radiative bands in molecular radiation are dependent uponvibrational temperature, free-free electron radiation is a function of electronic temperature, etc. There is alsoa linear dependence of the radiation upon the species densities in the flowfield - radiation, and its adsorptionbeing not only a function of wavelength, but also which species is affected by which wavelength throughquantum mechanics. This fact makes not only accurate species calculation important, but introduces anadditional degree of complexity in an ablative environment. Of particular concern is the effect of ablationwith carbon species ablation shields, since carbon is not only a strong absorber of radiation, but also a strongre-emitter of radiation.

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II. Flow Solver

For the purposes of this study, a seven-species air chemistry model with one ionized species was used. Theflow solver of Josyula and Bailey1 was selected due to its ability to independently track the vibrational energyof diatomic species in the flow. Knowledge of species specific vibrational energy allows for very precise energycalculations, particularly when considering radiative emission and absorption. The flow solver of Josyulaand Bailey is a finite volume, Roe approximate Riemann solver. The solver is second-order accurate throughthe employment of a MUSCL method, with a minmod limiter reducing the accuracy to first-order in thevicinity of strong shocks. The solver evaluates viscous flux terms using central differencing. A time-explicitpredictor-corrector method is used in the solver. The solver assumes that translational and rotational energymodes are in thermodynamic equilibrium, while independently tracking the vibrational energy mode of eachdiatomic species and the energy associated with the free electrons. The chemical species considered by thecode are: N2, O2, N , O, NO, NO+ and e−. A detailed discussion of the solver is presented in Ref 1. Abrief overview of solver’s development is presented below:

Species Mass ConservationThe species mass conservation is given by

∂

∂tρs +

∂

∂xjρsu

j =∂

∂xj(ρsV

js

)+ ω̇s (1)

where the diffusion velocity of species s is given by:

V js = ujs − uj (2)

The term ω̇s represents the rate of species creation and destruction, who’s sum over all species is zero.∑s

ω̇s = 0 (3)

Similarly, the sum of mass flux due to diffusion is also zero.∑s

ρsVjs = 0 (4)

The dissociation rates were determine using a functional form of the Arrhenius equation:

kf (Teff) = CfTηeffe

θd/Teff (5)

where the effective temperature, Teff is determined using the two-temperature model.

Teff =√TTv,s (6)

Total Momentum ConservationThe total momentum conservation is given by

∂

∂t(ρui) +

∂

∂xj(ρuiuj) +

∂P

∂xi− ∂τ ij

∂xj=∑s

NsesZsEi (7)

The right hand side of the preceding equation accounts for the electric effects upon the flowfield, withEi representing an approximation of the electric field.

Total Energy ConservationThe equation used to enforce the conservation of total energy is

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∂

∂t

[ρ

(12u2 + e

)]+

∂

∂xj

[ρuj

(12u2 + e

)]+∂qj

∂xj+

∂

∂xj(ujP )− ∂

∂xj(uiτ ij) =

∑s

εNsZsEiuis (8)

Vibrational Energy Conservation EquationThe vibrational energy of each diatomic species is determined by

∂

∂t(ρsev,s) +

∂

∂xj(ρsev,su

j)

=∂

∂xj

(η′v,s

∂Tv∂xj

)− ∂

∂xj(ρsev,sV

js

)+QT−V +Qe−V + ω̇sDs (9)

The solver’s treatment of vibrational energy in a manner that is specific to each species is unique, andwas a determining factor in its selection for this study. The species energy equations allow for the exactdetermination of the vibrational temperature of each species. It also allows for the potential inclusion of aradiative source term, which would be an entry point to coupling the flowfield with a radiative solver.

SPRADIAN2 was selected as the radiation code in this study. These two codes have been integratedtogether in the Coupled HYpersonic-Radiative Analysis (CHYRA) code.

III. Coupling Methodology

In the first phase of this study, the two codes have been integrated, but are run in an uncoupled fashion.A converged flowfield is generated by the flowfield portion of the code. Once the flow has converged, thespecies concentrations, heavy particle temperature, and the species vibrational temperatures are passed asinputs to the radiation portion of the code. For both geometries used in this study, a flowfield resolution of50 cells in the i-direction, and 60 cells in the j-direction was used.

To facilitate an accurate calculation of flow quantities at the shock-front and in the boundary layer, thegrid adaptation method of Gnoffo et. al.3 was used. Grid adaptation was executed multiple time duringthe course of a computational run. A trial-and-error process was used to balance the number of cells usedto refine the shock and boundary layers, with stability limits due to cell aspect ratio.

The spectral resolution used for the radiation calculation was driven by the need to compare radiative sur-face intensity with the FIRE-II vehicle. As in the FIRE experiment, the lower bound of radiation wavelengthwas driven by the window cutoff in the VUV region4 (2000 Angstrom). The coupled code was unstable,however, when radiation was considered below 3,000 Angstrom. An investigation is ongoing to determinethe source of the instability. The results presented in this paper will have a lower bound of 3,000 Angstrom.The upper wavelength bound was set at 12,000 Angstrom. A spectral resolution of 100,000 points was used.

Figure 1. Determination of the angular extent of eachneighbor.

Once the radiation portion of the code has de-termined the emission and absorption spectrumthroughout the flowfield, a radiation update is per-formed to determine radiative intensity at any pointin the flowfield and on all surfaces. A tangent-slabor spherical gas-cap model could be used to de-termine the surface intensities. Inaccuracies wouldarise however, for geometries and flowfields that vi-olated the geometric assumptions of these models.The primary assumption of concern is that the radi-ation environment is uniform with respect to angleas viewed from the surface.

A new radiation propagation algorithm is pre-sented which does not assume that the radiationfield is uniform with respect to angle. The algo-rithm is two-dimensional in nature, and does notinherently take into account the effects of the third-dimension. This may prove to be a complication forfurther integration. The basic steps in the methodare:

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1. Invoke SPRADIAN to determine spectral emission and absorption coefficients in each cell. SPRA-DIAN has been modified to allow species-specific vibrational temperatures to be used in the radiationcalculations.

2. Choose a number of transmission directions, and determine the angular extent of each direction. Ina traditional tangent-slab analysis, two transmission directions are considered; one towards the bodyand one away from the body. In this method, an arbitrary number of directions may be chosen, eachof which equally share a 2π two-dimensional angle. For instance, if four transmission directions werechosen, each direction would cover a π/2 angular region. The first direction is aligned with the +x axis.For the results published in this paper, 12 transmission directions were used, each direction covering aπ/6 included angle.

3. Cycling over domain, determine the angle formed by the ray from all cell centers to the edge midpointof each adjoining neighbor, as shown in figure 1. These angles are used to determine which angles ofradiation propagation are going to intercept each neighbor.

4. Cycling over each transmission direction, compare the minimum and maximum angles of the trans-mission direction to the minimum and maximum angles of the rays to the neighboring cell edges. Ifan overlap occurs, this indicates that emissions from the center cell in that transmission direction willintersect a given number of neighboring cells. The extent of this overlap is the ratio of the angularextent of the overlap (αdir) to the total included angle of the transmission direction (β). This coef-ficient is stored for future use. The dimensions needed to store this information are proportional tothe number of cells in the domain, the number of transmission directions chosen, and the number ofneighbors each cell has.

ci,j→in,jn =αdirβ

(10)

5. Cells at the edge of the shock and cells in the boundary layer have very high aspect ratios. Becauseof the aspect ratio, the angular extents of each neighbor vary greatly for each differential element ina cell. A differential volume on one side of a cell may transmit to a given neighbor over a wide rangeof angles, while a differential element on the other side of the cell may only transmit to that neighborover a very small range of angles. To accommodate this behavior, the transmission-neighbor overlapis calculated using five “center-points” equally distributed in the i-direction. Repeat the previous twoitems while moving the center-point used to generate the rays to neighboring cells along the i-axis.These coefficients are then averaged to determine an aggregate value for the entire cell.

6. Update the radiation field through a series of alternating sweeps (+i,+j) (−i,−j) (+j,+i) (−j,−i).In each increment of the sweep, accomplish the following:

(a) Initialize radiation intensity in wall-bounded cells using a grey-body distribution for all directionspointing away from the wall.

Iw,λ = 0.8Ibb,λ (11)

(b) Computationally examine each neighboring cell. For each neighboring cell, cycle over all transmis-sion directions. If the neighboring cell does not exist (edge of the domain), or if this transmissiondirection does not overlap this neighbor, then no update is made to the radiative intensity in theneighboring cell.

(c) If an overlap exists between a transmission direction and this neighbor, invoke SPRADIAN to solvethe radiation propagation equation5 (equation 12), assuming a liner variation of both emissioncoefficient and absorption coefficient. The intensity and emissivity of the originating cell are scaledby the proportion of overlap with this direction and this neighbor. For example, if a transmissiondirection equally overlaps two cells, only half of the emissivity and intensity is used to calculatean updated intensity in each neighbor, thereby enforcing conservation of energy. The distanceused the integration of the radiation propagation equation is the distance between cell centers.

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dIλds

= ε− κIλ (12)

The total intensity in the cell in each direction is determined by adding the contributions toradiative intensity in this direction from each neighboring cell.

Ii,j,λ,dir =∑in,jn

[Iλ,dir]in,jn→i,j (13)

7. Large arrays are required to store the spectral intensity information throughout the domain. Theintensity must be stored at each wavelength in each direction within every cell. The memory requiredis proportional to the number of cells in the domain multiplied by the product of the number oftransmission directions and the number of points in the spectrum.

8. To determine the radiative heat flux at the wall, the transmission direction that aligns the best withthe wall normal is selected. The intensity in this direction is integrated with respect to wavelength todetermine the intensity per Steradian [W/cm2 − sr]. For comparison with the FIRE-II experiment,the intensity per Steradian is multiplied by the field of view of the sensor (.023909 sr) which yields thelocal radiative heat flux [W/cm2].

IV. Results

A. FIRE-II

The FIRE-II vehicle was examined at the 1634s point in the trajectory. This point was selected as it is oneof the latest points in the first heat shield portion of the experiment. At this point, the vehicle was in thecontinuum flow regime with a relatively low Reynolds number. The wall of the FIRE-II vehicle was modeledas electrically catalytic, forcing the recombination of NO+ and e−. While it would be desirable to matchthe spectrum measured in the experiment (0.2µm to 4.0µm), this analysis considered radiation in the 0.3µmto 1.2µm wavelengths. These wavelengths were chosen for stability and resolution. A spectral resolution for100,000 points was used in both the tangent-slab analysis and the CHYRA analysis.

The grid used in FIRE-II runs is shown in figure 2. Grid alignment routines were able to to effectivelyadapt the grid to resolve both the boundary layer and the shock front. A summary of flow conditions ispresented in table 1.

Table 1. FIRE-II Conditions

Parameter ValueAltitude (km) 75.7

M∞ 40.1T∞ (K) 192Tw (K) 615

ρ∞ (kg/m3) 4.26× 10−5

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(a) Initial (b) Post-adaptation

Figure 2. FIRE-II grid 50x60

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1. Flowfield

(a) Mass Fractions

(b) Temperatures

Figure 3. Stagnation values for FIRE-II

The peak translational temperature behind the shock is justunder 40,000K. The vibrational temperatures of O2 and N2

equilibrate after a similar relaxation distance. The vibrationaltemperature of NO+ has a similar relaxation distance, howeverthere is a significant diffusion effect which increases the overallvibrational temperature of NO+ upstream of the shock. Thisdiffusion behavior is discussed in more detail in Ref 1.

Diatomic oxygen and nitrogen completely dissociate behindthe shock governed by the associated chemical rates. Once ni-trogen and oxygen atoms are present, NO forms, and is quicklyionized into NO+ and e−. The shock stand-off distance pre-dicted in this simulation is appreciably larger than the distancepredicted in other verification studies, such as Hash, et al.6

This is not surprising, given the more limited chemistry of aseven-species set.

2. Radiation Intensity

Table 2 shows the comparison of the predicted stagnation linesurface intensities on the FIRE-II vehicle using the presentedmethod and the traditional tangent slab method. Also shownon the table is the experimental value. CHYRA using thetangent-slab method slightly under-predicts the radiation in-tensity at the surface. This is not surprising due to the fact thatCHYRA does not include radiation from certain trace species.An additional contributor to the under-prediction of surfaceradiation is the absence of radiation in the 0.2µm to 0.3µmband. This band was omitted due to stability concerns, how-ever a significant portion of continuum O2 radiation is presentin this band.

CHYRA using the method presented above slightly over-predicts the radiation incident to the surface at the stagnationpoint, when compared to the tangent slab analyses. Note that neither of these predicted surface intensityresults account for the transmissivity of the radiometer transparency materiel. Accounting for this effect willfurther reduce the measured surface intensity in a way that brings the predictions closer to the experimentalresults. Figure 4 shows the new method predicting uniformly high average radiation intensities throughoutthe flowfield, when compared to the tangent slab analysis.

Table 2. FIRE-II stagnation pointsurface intensity measurements

Source Intensity (W/cm2)Experiment 8.40Tan-slab 7.40*

CHYRA 17.93*

* Does not account for transmissivity oftransparency material.

B. RAM-C

The RAM-C experiment was conducted in a weakly-ionized flow regime. Although the experiment did notmeasure radiation incident to the vehicle, this type of geometry contains features similar to other bodies ofinterest. For comparison with this case, the wall was modeled as partially catalytic. Equilibrium species

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(a) Tangent-Slab (b) CHRYA Method

Figure 4. Predicted average intensity in FIRE-II flowfield (W/cm2 − sr)

concentrations were enforced at the wall for air at 1,500K. Like the FIRE-II case, radiation was examinedin the 0.3µm to 1.2µm wavelengths. A spectral resolution of 100,000 points was use in the tangent-slabcalculation, while a resolution of 4,000 points was used in the CHYRA analysis, using the method previouslypresented. The 4,000 point resolution is insufficient to adequately resolve the spectral peaks, and does notrepresent a final result for this geometry.

The grid used in the RAM-C investigation is shown in figure 5. In this case, the grid alignment routinesnot only clustered the cells near the shock front and boundary layer, but also positively affected the gridmetrics by promoting the orthogonality of the grid. A summary of flow conditions is presented in table 3.

(a) Initial (b) Post-adaptation

Figure 5. RAM-C grid 50x60

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Table 3. RAM-C Conditions

Parameter ValueAltitude (km) 61

M∞ 23.9T∞ (K) 254Tw (K) 1,500

ρ∞ (kg/m3) 2.69× 10−4

1. Flowfield

(a) Mass Fractions

(b) Temperatures

Figure 6. Stagnation values for RAM-C

The peak translational temperature behind the shock is ap-proximately 19,000K. The vibrational temperatures of O2 andN2 equilibrate after a similar relaxation distance. The vibra-tional temperature of NO+ has a similar relaxation distance,however there is a significant diffusion effect which increasesthe overall vibrational temperature of NO+ upstream of theshock.

Diatomic oxygen is completely dissociated behind theshock, while diatomic nitrogen is only partially dissociated.Low temperatures in the post-shock environment along withlow availability of oxygen leads to small concentrations of NO,NO+, and e−.

The RAM-C flowfield downstream of the blunt nose experi-ences large changes in temperatures and species concentrations,as shown in figure 7. This region of the flow presents a verydifferent environment in which ablative products (if modeled)could convect and radiate in previously unexpected ways.

2. Radiation Intensity

Due to the poor spectral resolution of the RAM-C simulationusing the CHYRA method, we are unable to present compar-isons of flowfield radiation intensities. The tangent slab methodpredicts very low stagnation point radiation intensity, as shownin table 4. This intensity was determined by integrating overthe same solid angle as the FIRE-II experiment.

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Figure 7. RAM-C downstream variation of NO concentration.

Table 4. RAM-C stagnation point surface intensity measurements

Source Intensity (W/cm2)Tan-slab 1.4× 10−2

CHYRA N/A

V. Future Work

Future development on the Coupled HYpersonic-Radiative Analysis (CHYRA) software will be focusedin the following areas:

1. Refinement of 2-D radiation propagation algorithm. The current implementation of the algorithmshows reasonable agreement with the surface intensities measured in the FIRE-II experiment, andsurface intensities

2. Direct coupling of radiation emission and absorption into the species energy equations. Difficultiesin matching radiative intensity at the wall in the FIRE-II case with experimental values has delayedefforts to further integrate the flowfield portions of the code with the radiation portions of the code.Once the current issues are resolved, species specific emission can be accounted for in each cell, alongwith species specific absorption.

3. Increase the number of species tracked by the code. The seven-species model currently in use misses theradiation contributions due to several trace species, such as N+

2 . Although these species do not accountfor a significant portion of the gas’ mass, many of the species do provide significant contributions tothe radiation environment.7 An 11-species model is desired for implementation initially, followed by amodel that includes multiple carbon species to model radiation due to ablation by-products.

4. Inclusion of an ablative wall condition. The eventual capability desired from this tool requires thatphysically accurate ablation processes and mass injection are modeled at the wall.

VI. Conclusions

Coupled solutions for the hypersonic flowfield and radiation environment are necessary for robust designof future hypersonic vehicles. A multi-species hypersonic code that individually tracks species vibrationalenergies is a promising method for highly accurate temperature calculation, which is required for radiation

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prediction. The current hypersonic flow solver requires further development to extend the chemistry set, andincrease the stability of the code for arbitrary geometries and flight conditions.

Independently tracking radiation intensity in an arbitrary number of transmission directions was demon-strated. The current implementation of the multi-direction, spectrally accurate radiation tracking did notshow agreement with tangent-slab calculations, or experiment. Future work will be conducted to refine themethod and increase the accuracy of determining radiation intensity.

References

1Josyula, E. and Bailey, W. F., “Governing Equations for Weakly Ionized Plasma Flowfields of Aerospace Vehicles,” Journalof Spacecraft and Rockets, Vol. 40, No. 6, November-December 2003, pp. 845–857.

2Fujita, K. and Abe, T., “SPRADIAN, Structured Package for Radiation Analysis: Theory and Application,” Report 669,The Institute of Space and Astronautical Science, September 1997.

3Gnoffo, P. A., Hartung, L. C., and Greendyke, R. B., “Heating Analysis for a Lunar Transfer Vehicle at Near-EquilibriumFlow Conditions,” 93-0270, 31st AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, January 1993.

4Cauchon, D. L., “Radiative Heating Results from the FIRE II Flight Experiment at a Reentry Velocity of 11.4 Kilometersper Second,” NASA TM X-1402, National Air and Space Administration, April 1972.

5Park, C., Nonequilibrium Hypersonic Aerothermodynamics, John Wiley & Sons, New York, 1990.6Hash, D., Olejniczak, J., Wright, M., Prabhu, D., Pulsonetti, M., Hollis, B., Gnoffo, P., Barnhardt, M., Nompelis, I., and

Candler, G., “FIRE II Calculation for Hypersonic Nonequilibrium Aerothermodynamics Code Verification: DPLR, LAURA,and US3D,” 2007-605, 45th AIAA Aerospace Sciences Meeting, Reno, Nevada, 2007.

7Greendyke, R. B., “A Parametric Analysis of Radiative Structure in Aerobrake Shock Layers,” 92-2970, AIAA 23rdPlasmadynamics & Lasers Conference, Nashville, TN, July 1992.

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