# ASTM E637-05 (Reapproved 2016)

Designation: E637 − 05 (Reapproved 2016)Standard Test Method forCalculation of Stagnation Enthalpy from Heat TransferTheory and Experimental Measurements of Stagnation-PointHeat Transfer and Pressure1This standard is issued under the fixed designation E637; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon (´) indicates an editorial change since the last revision or reapproval.INTRODUCTIONThe enthalpy (energy per unit mass) determination in a hot gas aerodynamic simulation device isa difficult measurement. Even at temperatures that can be measured with thermocouples, there aremany corrections to be made at 600 K and above. Methods that are used for temperatures above therange of thermocouples that give bulk or average enthalpy values are energy balance (see PracticeE341), sonic flow (1, 2),2and the pressure rise method (3). Local enthalpy values (thus distribution)may be obtained by using either an energy balance probe (see Method E470), or the spectrometrictechnique described in Ref (4).1. Scope1.1 This test method covers the calculation from heattransfer theory of the stagnation enthalpy from experimentalmeasurements of the stagnation-point heat transfer and stagna-tion pressure.1.2 Advantages:1.2.1 A value of stagnation enthalpy can be obtained at thelocation in the stream where the model is tested. This valuegives a consistent set of data, along with heat transfer andstagnation pressure, for ablation computations.1.2.2 This computation of stagnation enthalpy does notrequire the measurement of any arc heater parameters.1.3 Limitations and Considerations—There are many fac-tors that may contribute to an error using this type of approachto calculate stagnation enthalpy, including:1.3.1 Turbulence—The turbulence generated by adding en-ergy to the stream may cause deviation from the laminarequilibrium heat transfer theory.1.3.2 Equilibrium, Nonequilibrium, or Frozen State ofGas—The reaction rates and expansions may be such that thegas is far from thermodynamic equilibrium.1.3.3 Noncatalytic Effects—The surface recombination ratesand the characteristics of the metallic calorimeter may give aheat transfer deviation from the equilibrium theory.1.3.4 Free Electric Currents—The arc-heated gas streammay have free electric currents that will contribute to measuredexperimental heat transfer rates.1.3.5 Nonuniform Pressure Profile—A nonuniform pressureprofile in the region of the stream at the point of the heattransfer measurement could distort the stagnation point veloc-ity gradient.1.3.6 Mach Number Effects—The nondimensionalstagnation-point velocity gradient is a function of the Machnumber. In addition, the Mach number is a function of enthalpyand pressure such that an iterative process is necessary.1.3.7 Model Shape—The nondimensional stagnation-pointvelocity gradient is a function of model shape.1.3.8 Radiation Effects—The hot gas stream may contributea radiative component to the heat transfer rate.1.3.9 Heat Transfer Rate Measurement—An error may bemade in the heat transfer measurement (see Method E469 andTest Methods E422, E457, E459, and E511).1.3.10 Contamination—The electrode material may be of alarge enough percentage of the mass flow rate to contribute tothe heat transfer rate measurement.1.4 The values stated in SI units are to be regarded asstandard. No other units of measurement are included in thisstandard.1.4.1 Exception—The values given in parentheses are forinformation only.1.5 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is the1This test method is under the jurisdiction of ASTM Committee E21 on SpaceSimulation and Applications of Space Technology and is the direct responsibility ofSubcommittee E21.08 on Thermal Protection.Current edition approved April 1, 2016. Published April 2016. Originallyapproved in 1978. Last previous edition approved in 2011 as E637 – 05 (2011).DOI: 10.1520/E0637-05R16.2The boldface numbers in parentheses refer to the list of references appended tothis method.Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States1responsibility of the user of this standard to establish appro-priate safety and health practices and determine the applica-bility of regulatory limitations prior to use.2. Referenced Documents2.1 ASTM Standards:3E341 Practice for Measuring Plasma Arc Gas Enthalpy byEnergy BalanceE422 Test Method for Measuring Heat Flux Using a Water-Cooled CalorimeterE457 Test Method for Measuring Heat-Transfer Rate Usinga Thermal Capacitance (Slug) CalorimeterE459 Test Method for Measuring Heat Transfer Rate Usinga Thin-Skin CalorimeterE469 Measuring Heat Flux Using a Multiple-Wafer Calo-rimeter (Withdrawn 1982)4E470 Measuring Gas Enthalpy Using Calorimeter Probes(Withdrawn 1982)4E511 Test Method for Measuring Heat Flux Using a Copper-Constantan Circular Foil, Heat-Flux Transducer3. Significance and Use3.1 The purpose of this test method is to provide a standardcalculation of the stagnation enthalpy of an aerodynamicsimulation device using the heat transfer theory and measuredvalues of stagnation point heat transfer and pressure. Astagnation enthalpy obtained by this test method gives aconsistent set of data, along with heat transfer and stagnationpressure for ablation computations.4. Enthalpy Computations4.1 This method of calculating the stagnation enthalpy isbased on experimentally measured values of the stagnation-point heat transfer rate and pressure distribution and theoreticalcalculation of laminar equilibrium catalytic stagnation-pointheat transfer on a hemispherical body. The equilibrium cata-lytic theoretical laminar stagnation-point heat transfer rate fora hemispherical body is as follows (5):qŒRPt25 Ki~He2 Hw! (1)where:q = stagnation-point heat transfer rate, W/m2(or Btu/ft2·s),Pt2= model stagnation pressure, Pa (or atm),R = hemispherical nose radius, m (or ft),He= stagnation enthalpy, J/kg (or Btu/lb),Hw= wall enthalpy, J/kg (or Btu/lb), andKi= heat transfer computation constant.4.2 Low Mach Number Correction—Eq 1 is simple andconvenient to use since Kican be considered approximatelyconstant (see Table 1). However, Eq 1 is based on a stagnation-point velocity gradient derived using “modified” Newtonianflow theory which becomes inaccurate for Moo0.1where:β = stagnation-point velocity gradient, s−1,D = hemispherical diameter, m (or ft),U∞= freestream velocity, m/s (or ft/s),(βD/U∞)x=0= dimensionless stagnation velocity gradient,KM= enthalpy computation constant,(N1/2·m1/2· s)/kg or (ft3/2·atm1/2·s)/lb, andM∞ = the freestream Mach number.For subsonic Mach numbers, an expression for (βD/U∞)x=0for a hemisphere is given in Ref (6) as follows:SβDU`Dx505 3 2 0.755 M`2~M`,1! (4)For a Mach number of 1 or greater, (βD/U∞)x=0for ahemisphere based on “classical” Newtonian flow theory ispresented in Ref (7) as follows:SβDU`Dx50558@~γ 2 1!M`2 12#~γ11!M`2311γ 2 12@~γ 2 1!M`2 12#2γM`2 2 ~γ 2 1!421γ2160.5(5)Avariation of (βD/U∞)x=0with M∞and γ is shown in Fig. 1.The value of the Newtonian dimensionless velocity gradientapproaches a constant value as the Mach number approachesinfinity:SβDU`Dx50,M→`5 Œ4Sγ 2 1γD(6)and thus, since γ, the ratio of specific heats, is a function ofenthalpy, (βD/U∞)x=0is also a function of enthalpy. Again, aniteration is necessary. From Fig. 1, it can be seen that3For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at service@astm.org. For Annual Book of ASTMStandards volume information, refer to the standard’s Document Summary page onthe ASTM website.4The last approved version of this historical standard is referenced onwww.astm.org.TABLE 1 Heat Transfer and Enthalpy Computation Constants forVarious GasesGasKi, kg/(N1/2·m1/2·s)(lb/(ft3/2·s·atm1/2))KM,(N1/2·m1/2·s)/kg((ft3/2·s·atm1/2)/lb)Air 3.905 × 10−4(0.0461) 2561 (21.69)Argon 5.513 × 10−4(0.0651) 1814 (15.36)Carbon dioxide 4.337 × 10−4(0.0512) 2306 (19.53)Hydrogen 1.287 × 10−4(0.0152) 7768 (65.78)Nitrogen 3.650 × 10−4(0.0431) 2740 (23.20)E637 − 05 (2016)2(βD/U∞)x =0for a hemisphere is approximately 1 for largeMach numbers and γ = 1.2. KMis tabulated in Table 1 using(βD/U∞)x =0= 1 and Kifrom Ref (5).4.3 Mach Number Determination:4.3.1 The Mach number of a stream is a function of the totalenthalpy, the ratio of freestream pressure to the total pressure,p/pt1, the total pressure, pt1, and the ratio of the exit nozzle areato the area of the nozzle throat, A/A .Fig. 2(a) and Fig. 2(b) arereproduced from Ref (8) for the reader’s convenience indetermining Mach numbers for supersonic flows.4.3.2 The subsonic Mach number may be determined fromFig. 3 (see also Test Method E511).An iteration is necessary todetermine the Mach number since the ratio of specific heats, γ,is also a function of enthalpy and pressure.FIG. 1 Dimensionless Velocity Gradient as a Function of Mach Number and Ratio of Specific HeatsFIG. 2 (a) Variation of Area Ratio with Mach NumbersE637 − 05 (2016)3FIG. 2 (b) Variation of Area Ratio with Mach Numbers (continued)FIG. 3 Subsonic Pressure Ratio as a Function of Mach Number and γE637 − 05 (2016)44.3.3 The ratio of specific heats, γ, is shown as a function ofentropy and enthalpy for air in Fig. 4 from Ref (9). S/R is thedimensionless entropy, and H/RT is the dimensionless en-thalpy.4.4 Velocity Gradient Calculation from PressureDistribution—The dimensionless stagnation-point velocitygradient may be obtained from an experimentally measuredpressure distribution by using Bernoulli’s compressible flowequation as follows:SUU`D5@1 2 ~p/pt2!γ21γ #0.5@1 2 ~p`/pt2!γ21γ #0.5(7)where the velocity ratio may be calculated along the bodyfrom the stagnation point. Thus, the dimensionless stagnation-point velocity gradient, (βD/U∞)x=0, is the slope of the U/U∞and the x/D curve at the stagnation point.4.5 Model Shape—The nondimensional stagnation-point ve-locity gradient is a function of the model shape and the Machnumber. For supersonic Mach numbers, the heat transferrelationship between a hemisphere and other axisymmetricblunt bodies is shown in Fig. 5 (10).InFig. 5, rcis the cornerradius, rbis the body radius, rnis the nose radius, and q˙s,his thestagnation-point heat transfer rate on a hemisphere. For sub-sonic Mach numbers, the same type of variation is shown inFig. 6(6).4.6 Radiation Effects:4.6.1 As this test method depends on the accurate determi-nation of the convective stagnation-point heat transfer, anyradiant energy absorbed by the calorimeter surface and incor-rectly attributed to the convective mode will directly affect theoverall accuracy of the test method. Generally, the sources ofradiant energy are the hot gas stream itself or the gas heatingdevice, or both. For instance, arc heaters operated at highpressure (10 atm or higher) can produce significant radiantfluxes at the nozzle exit plane.4.6.2 The proper application requires some knowledge ofthe radiant environment in the stream at the desired operatingconditions. Usually, it is necessary to measure the radiant heattransfer rate either directly or indirectly. The following is a listof suggested methods by which the necessary measurementscan be made.4.6.2.1 Direct Measurement with Radiometer—Radiometersare available for the measurement of the incident radiant fluxwhile excluding the convective heat transfer. In its simplestform, the radiometer is a slug, thin-skin, or circular foilcalorimeter with a sensing area with a coating of knownabsorptance and covered with some form of window. Thepurpose of the window is to prevent convective heat transferfrom affecting the calorimeter while transmitting the radiantenergy. The window is usually made of quartz or sapphire. Thesensing surface is at the stagnation point of a test probe and islocated in such a manner that the view angle is not restricted.The basic radiometer view angle should be 120° or greater.This technique allows for immersion of the radiometer in thetest stream and direct measurement of the radiant heat transferrate. There is a major limitation to this technique, however, inthat even with high-pressure water cooling of the radiometerenclosure, the window is poorly cooled and thus the use ofwindows is limited to relatively low convective heat transferconditions or very short exposure times, or both. Also, streamcontaminants coat the window and reduce its transmittance.4.6.2.2 Direct Measurement with Radiometer Mounted inCavity—The two limitations noted in 4.6.2.1 may be overcomeby mounting the radiometer at the bottom of a cavity open tothe stagnation point of the test probe (see Fig. 7). Good resultscan be obtained by using a simple calorimeter in place of theradiometer with a material of known absorptance. When usingthis configuration, the measured radiant heat transfer rate isused in the following equation to determine the stagnation-point radiant heat transfer, assuming diffuse radiation:FIG. 4 Isentropic Exponent for Air in EquilibriumE637 − 05 (2016)5FIG. 5 Stagnation-Point Heating-Rate Parameters on Hemispherical Segments of Different Curvatures for Varying Corner-Radius RatiosE637 − 05 (2016)6q˙r151α2F12q˙r2(8)where:q˙r1= radiant transfer at stagnation point,q˙r2= radiant transfer at bottom of cavity (measured),α2= absorptance of sensor surface, andF12= configuration factor.For a circular cavity geometry (recommended), F12isConfiguration A-3 of Ref (11)and can be determined from thefollowing equation:F125 1/2 @X 2 ~X22 4E2D2!1/2# (9)FIG. 6 Stagnation-Point Heat Transfer Ratio to a Blunt Body and a Hemisphere as a Function of theBody-to-Nose Radius in a Subsonic StreamFIG. 7 Test ProbeE637 − 05 (2016)7where:E = r2/d,D = d/r1,X = 1+(1+E2)D2, andr1, d, and r2are defined in Fig. 8.The major limitation of this particular technique is due toheating of the cavity opening (at the stagnation point). If thetest probe is inadequately cooled or uncooled, heating at thispoint can contribute to the radiant heat transfer measured at thesensor and produce large errors. This method of measuring theradiant heat transfer is then limited to test conditions and probeconfigurations that allow for cooling of the probe in thestagnation area such that the cavity opening is maintained at atemperature less than about 700 K.4.6.2.3 Indirect Measurement—At the highest convectiveheating rates, the accurate determination of the radiant fluxlevels is difficult. There are many schemes that could be usedto measure incident radiant flux indirectly. One such would bethe measurement of the radiant flux reflected from a surface inthe test stream. This technique depends primarily on theaccurate determination of surface reflectance under actual testconditions. The surface absorptance and a measurement of thesurface temperature at the point viewed by the radiant fluxmeasuring device ar