Tank Test

By using data from a tank test output, it is possible to obtain the temperature and the mass flow of the gas supplied, which can be used as input to Radioss.

With a tank test it is possible to measure the pressure at the injection point or in the middle of the tank, the two values are equal so the pressure variation is well known. Also, the amount of gas supplied and the characteristics of the gas in the test are known.

For temperature, it is more difficult to get accurate test data because the temperature gauges are often not accurate enough. Therefore, the temperature from the tank test may be wrong.


Figure 1. Unknown Mass Flow Rate

The following cases take into account that you may or may not know the injected temperature and the temperature in the tank.

In case the temperature at the injector and in the tank is not known:
  • Initial and injected gas composition
    Knowing the molecular weight ( M W i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbGaam 4vamaaBaaaleaacaWGPbaabeaaaaa@3927@ ) of each elements and the molar fraction ( X i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGybWaaS baaSqaaiaadMgaaeqaaaaa@3856@ ), it is possible to define the molar weight of the gas ( M W MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbGaam 4vaaaa@380D@ ):(1)
    MW= X i M W i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbGaam 4vaiabg2da9maaqaeabaGaamiwamaaBaaaleaacaWGPbaabeaakiab gwSixlaad2eacaWGxbWaaSbaaSqaaiaadMgaaeqaaaqabeqaniabgg HiLdaaaa@422F@
    The average heat capacity, per mass unit of a mixture of gases is given by the Amagat-Leduc equation:(2)
    C p ( T )= m i C pa m i + m i C pb m i T+ m i C pc m i T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGdbWaaS baaSqaaiaadchaaeqaaOWaaeWaaeaacaWGubaacaGLOaGaayzkaaGa eyypa0ZaaSaaaeaadaaeabqaaiaad2gadaWgaaWcbaGaamyAaaqaba GccqGHflY1caWGdbWaaSbaaSqaaiaadchacaWGHbaabeaaaeqabeqd cqGHris5aaGcbaWaaabqaeaacaWGTbWaaSbaaSqaaiaadMgaaeqaaa qabeqaniabggHiLdaaaOGaey4kaSYaaSaaaeaadaaeabqaaiaad2ga daWgaaWcbaGaamyAaaqabaGccqGHflY1caWGdbWaaSbaaSqaaiaadc hacaWGIbaabeaaaeqabeqdcqGHris5aaGcbaWaaabqaeaacaWGTbWa aSbaaSqaaiaadMgaaeqaaaqabeqaniabggHiLdaaaOGaamivaiabgU caRmaalaaabaWaaabqaeaacaWGTbWaaSbaaSqaaiaadMgaaeqaaOGa eyyXICTaam4qamaaBaaaleaacaWGWbGaam4yaaqabaaabeqab0Gaey yeIuoaaOqaamaaqaeabaGaamyBamaaBaaaleaacaWGPbaabeaaaeqa beqdcqGHris5aaaakiaadsfadaahaaWcbeqaaiaaikdaaaaaaa@686E@

Using the previous equation, the heat capacity coefficients ( C p ( T ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaWGWbaabeaakiaacIcacaWGubGaaiykaaaa@3A1B@ ) of the initial and injected mixture may be defined.

Knowing the characteristics of the injected gas, the initial gas, and the mixture, it is possible to find the mass flow and the temperature for the inflator. The following basic equations are used to carry out the analysis.
  • The perfect gas equation of state is:(3)
    P V = n R T , n = m M W
    with R = 8.314 J m o l e K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGsbGaey ypa0JaaGioaiaac6cacaaIZaGaaGymaiaaisdadaWcaaqaaiaacQea aeaacaGGTbGaai4BaiaacYgacaGGLbGaeyyXICTaai4saaaaaaa@4399@
  • And the adiabatic equation:(4)
    H = c o n s t . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibGaey ypa0Jaam4yaiaad+gacaWGUbGaam4CaiaadshacaGGUaaaaa@3DA3@
    with H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibaaaa@372B@ being the total enthalpy of the system (inflator + tank).
From the conservation of the energy, the basic energy equation of the tank test can be written as:(5)
d E airbag =PdV+d H in d H out MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGKbGaam yramaaBaaaleaacaWGHbGaamyAaiaadkhacaWGIbGaamyyaiaadEga aeqaaOGaeyypa0JaeyOeI0IaamiuaiaadsgacaWGwbGaey4kaSIaam izaiaadIeadaWgaaWcbaGaamyAaiaad6gaaeqaaOGaeyOeI0Iaamiz aiaadIeadaWgaaWcbaGaam4BaiaadwhacaWG0baabeaaaaa@4CBC@

Here d H o u t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGKbGaam isamaaBaaaleaacaWGVbGaamyDaiaadshaaeqaaaaa@3B27@ =0, since the tank test is adiabatic. The constant volume of the tank test means d V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGKbGaam Ovaaaa@3822@ =0.

Therefore, Equation 5 is summarized as:(6)
U f U 0 = d H i n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaS baaSqaaiaadAgaaeqaaOGaeyOeI0IaamyvamaaBaaaleaacaaIWaaa beaakiabg2da9iaadsgacaWGibWaaSbaaSqaaiaadMgacaWGUbaabe aaaaa@3FD9@

0 T mix m ( in+air ) C V( in+air ) dT 0 T air m ( air ) C V( air ) dT= 0 T in m ( in ) C P( in ) dT MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHuhY2da WdXbqaaiaad2gadaWgaaWcbaWaaeWaaeaacaWGPbGaamOBaiabgUca RiaadggacaWGPbGaamOCaaGaayjkaiaawMcaaaqabaaabaGaaGimaa qaaiaadsfadaWgaaadbaGaamyBaiaadMgacaWG4baabeaaa0Gaey4k IipakiabgwSixlaadoeadaWgaaWcbaGaamOvamaabmaabaGaamyAai aad6gacqGHRaWkcaWGHbGaamyAaiaadkhaaiaawIcacaGLPaaaaeqa aOGaamizaiaadsfacqGHsisldaWdXbqaaiaad2gadaWgaaWcbaWaae WaaeaacaWGHbGaamyAaiaadkhaaiaawIcacaGLPaaaaeqaaaqaaiaa icdaaeaacaWGubWaaSbaaWqaaiaadggacaWGPbGaamOCaaqabaaani abgUIiYdGccqGHflY1caWGdbWaaSbaaSqaaiaadAfadaqadaqaaiaa dggacaWGPbGaamOCaaGaayjkaiaawMcaaaqabaGccaWGKbGaamivai abg2da9maapehabaGaamyBamaaBaaaleaadaqadaqaaiaadMgacaWG UbaacaGLOaGaayzkaaaabeaaaeaacaaIWaaabaGaamivamaaBaaame aacaWGPbGaamOBaaqabaaaniabgUIiYdGccqGHflY1caWGdbWaaSba aSqaaiaadcfadaqadaqaaiaadMgacaWGUbaacaGLOaGaayzkaaaabe aakiaadsgacaWGubaaaa@815B@

m ( mix ) T mix ( C pa( mix ) + C pb( mix ) T mix 2 + C pc( mix ) T mix 2 3 R M W mix ) m ( air ) T 0 ( C pa( air ) + C pb( air ) T 0 2 + C pc( air ) T 0 2 3 R M W air )= m ( in ) T in ( C pa( in ) + C pb( in ) T in 2 + C pc( in ) T in 2 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaiabgs DiBlaad2gadaWgaaWcbaWaaeWaaeaacaWGTbGaamyAaiaadIhaaiaa wIcacaGLPaaaaeqaaOGaamivamaaBaaaleaacaWGTbGaamyAaiaadI haaeqaaOWaaeWaaeaacaWGdbWaaSbaaSqaaiaadchacaWGHbWaaeWa aeaacaWGTbGaamyAaiaadIhaaiaawIcacaGLPaaaaeqaaOGaey4kaS Iaam4qamaaBaaaleaacaWGWbGaamOyamaabmaabaGaamyBaiaadMga caWG4baacaGLOaGaayzkaaaabeaakmaalaaabaGaamivamaaBaaale aacaWGTbGaamyAaiaadIhaaeqaaaGcbaGaaGOmaaaacqGHRaWkcaWG dbWaaSbaaSqaaiaadchacaWGJbWaaeWaaeaacaWGTbGaamyAaiaadI haaiaawIcacaGLPaaaaeqaaOWaaSaaaeaacaWGubWaaSbaaSqaaiaa d2gacaWGPbGaamiEaaqabaGcdaahaaWcbeqaaiaaikdaaaaakeaaca aIZaaaaiabgkHiTmaalaaabaGaamOuaaqaaiaad2eacaWGxbWaaSba aSqaaiaad2gacaWGPbGaamiEaaqabaaaaaGccaGLOaGaayzkaaGaey OeI0IaamyBamaaBaaaleaadaqadaqaaiaadggacaWGPbGaamOCaaGa ayjkaiaawMcaaaqabaGccaWGubWaaSbaaSqaaiaaicdaaeqaaaGcba WaaeWaaeaacaWGdbWaaSbaaSqaaiaadchacaWGHbWaaeWaaeaacaWG HbGaamyAaiaadkhaaiaawIcacaGLPaaaaeqaaOGaey4kaSIaam4qam aaBaaaleaacaWGWbGaamOyamaabmaabaGaamyyaiaadMgacaWGYbaa caGLOaGaayzkaaaabeaakmaalaaabaGaamivamaaBaaaleaacaaIWa aabeaaaOqaaiaaikdaaaGaey4kaSIaam4qamaaBaaaleaacaWGWbGa am4yamaabmaabaGaamyyaiaadMgacaWGYbaacaGLOaGaayzkaaaabe aakmaalaaabaGaamivamaaBaaaleaacaaIWaaabeaakmaaCaaaleqa baGaaGOmaaaaaOqaaiaaiodaaaGaeyOeI0YaaSaaaeaacaWGsbaaba GaamytaiaadEfadaWgaaWcbaGaamyyaiaadMgacaWGYbaabeaaaaaa kiaawIcacaGLPaaacqGH9aqpcaWGTbWaaSbaaSqaamaabmaabaGaam yAaiaad6gaaiaawIcacaGLPaaaaeqaaOGaamivamaaBaaaleaacaWG PbGaamOBaaqabaGcdaqadaqaaiaadoeadaWgaaWcbaGaamiCaiaadg gadaqadaqaaiaadMgacaWGUbaacaGLOaGaayzkaaaabeaakiabgUca RiaadoeadaWgaaWcbaGaamiCaiaadkgadaqadaqaaiaadMgacaWGUb aacaGLOaGaayzkaaaabeaakmaalaaabaGaamivamaaBaaaleaacaWG PbGaamOBaaqabaaakeaacaaIYaaaaiabgUcaRiaadoeadaWgaaWcba GaamiCaiaadogadaqadaqaaiaadMgacaWGUbaacaGLOaGaayzkaaaa beaakmaalaaabaGaamivamaaBaaaleaacaWGPbGaamOBaaqabaGcda ahaaWcbeqaaiaaikdaaaaakeaacaaIZaaaaaGaayjkaiaawMcaaaaa aa@C0F8@

Inflator Temperature

In Equation 6, the unknown variable is only T i n i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaWGPbGaamOBaiaadMgaaeqaaaaa@39CA@ .

The other variables are known or could be determined using Equation 2 and Equation 3:
  • m ( i n ) m ( a i r ) = M W a i r P 0 V R T 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGTbWaaS baaSqaamaabmaabaGaamyAaiaad6gaaiaawIcacaGLPaaaaeqaaOGa amyBamaaBaaaleaadaqadaqaaiaadggacaWGPbGaamOCaaGaayjkai aawMcaaaqabaGccqGH9aqpcaWGnbGaam4vamaaBaaaleaacaWGHbGa amyAaiaadkhaaeqaaOWaaSaaaeaacaWGqbWaaSbaaSqaaiaaicdaae qaaOGaamOvaaqaaiaadkfacaWGubWaaSbaaSqaaiaaicdaaeqaaaaa aaa@4B67@ and n ( m i x ) = m ( i n ) M W i n + m ( a i r ) M W a i r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGUbWaaS baaSqaamaabmaabaGaamyBaiaadMgacaWG4baacaGLOaGaayzkaaaa beaakiabg2da9maalaaabaGaamyBamaaBaaaleaadaqadaqaaiaadM gacaWGUbaacaGLOaGaayzkaaaabeaaaOqaaiaad2eacaWGxbWaaSba aSqaaiaadMgacaWGUbaabeaaaaGccqGHRaWkdaWcaaqaaiaad2gada WgaaWcbaWaaeWaaeaacaWGHbGaamyAaiaadkhaaiaawIcacaGLPaaa aeqaaaGcbaGaamytaiaadEfadaWgaaWcbaGaamyyaiaadMgacaWGYb aabeaaaaaaaa@506D@
  • for i = a to c: C p i ( i n ) , C p i ( a i r ) , C p i ( m i x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGdbWaaS baaSqaaiaadchacaWGPbaabeaakmaaBaaaleaadaqadaqaaiaadMga caWGUbaacaGLOaGaayzkaaaabeaakiaacYcacaWGdbWaaSbaaSqaai aadchacaWGPbaabeaakmaaBaaaleaadaqadaqaaiaadggacaWGPbGa amOCaaGaayjkaiaawMcaaaqabaGccaGGSaGaam4qamaaBaaaleaaca WGWbGaamyAaaqabaGcdaWgaaWcbaWaaeWaaeaacaWGTbGaamyAaiaa dIhaaiaawIcacaGLPaaaaeqaaaaa@4D1D@ are calculated with Equation 2, and
  • T m i x = P t a n k V t a n k n ( m i x ) R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGubWaaS baaSqaaiaad2gacaWGPbGaamiEaaqabaGccqGH9aqpdaWcaaqaaiaa dcfadaWgaaWcbaGaamiDaiaadggacaWGUbGaam4AaaqabaGccaWGwb WaaSbaaSqaaiaadshacaWGHbGaamOBaiaadUgaaeqaaaGcbaGaamOB amaaBaaaleaacaGGOaGaamyBaiaadMgacaWG4bGaaiykaaqabaGcca WGsbaaaaaa@4B36@

Therefore, Equation 6 finds the temperature at the injector T i n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaWGPbGaamOBaiaadMgaaeqaaaaa@39CA@ of the injected gas by iterating on T i n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaWGPbGaamOBaiaadMgaaeqaaaaa@39CA@ . First, the temperature is guessed and six iterations are sufficient to converge to the solution.

Mass Flow

Knowing the evolution of the pressure versus time at the top of the tank test, it is possible to determine the mass flow rate with:(7)
m ˙ = m P P t ΔM ΔP P t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGTbGbai aacqGH9aqpdaWcaaqaaiabgkGi2kaad2gaaeaacqGHciITcaWGqbaa amaalaaabaGaeyOaIyRaamiuaaqaaiabgkGi2kaadshaaaGaeyyrIa 0aaSaaaeaacqqHuoarcaWGnbaabaGaeuiLdqKaamiuaaaadaWcaaqa aiabgkGi2kaadcfaaeaacqGHciITcaWG0baaaaaa@4C0C@
With,
Δ P
Total pressure variation during the experiment
Δ M
Total injected mass
Equation 7 may be written if the variation of mass versus the variation of the pressure is a function strictly growing, which is the case.


Figure 2. Mass Flow Curve - Evolution of the Mass Flow versus Time

Inflator Gas Velocity

Since the pressure is quickly uniform, the following equation may be written: P i n = P ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbWaaS baaSqaaiaadMgacaWGUbaabeaakiabg2da9iaadcfacaGGOaGaamiD aiaacMcaaaa@3D77@ , knowing T i n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaWGPbGaamOBaiaadMgaaeqaaaaa@39CA@ the density may be expressed as a function of T i n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaWGPbGaamOBaiaadMgaaeqaaaaa@39CA@ and P ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbGaai ikaiaadshacaGGPaaaaa@3985@ :(8)
ρ in = P(t) T in R M W in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHbpGCda WgaaWcbaGaamyAaiaad6gaaeqaaOGaeyypa0ZaaSaaaeaacaWGqbGa aiikaiaadshacaGGPaaabaWaaSGaaeaacaWGubWaaSbaaSqaaiaadM gacaWGUbaabeaakiaadkfaaeaacaWGnbGaam4vamaaBaaaleaacaWG PbGaamOBaaqabaaaaaaaaaa@4606@
Besides, V i n ( t ) = d m ( t ) d t S ρ i n ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGwbWaaS baaSqaaiaadMgacaWGUbaabeaakiaacIcacaWG0bGaaiykaiabg2da 9maalaaabaWaaSGaaeaacaWGKbGaamyBaiaacIcacaWG0bGaaiykaa qaaiaadsgacaWG0baaaaqaaiaadofacqaHbpGCdaWgaaWcbaGaamyA aiaad6gaaeqaaOGaaiikaiaadshacaGGPaaaaaaa@49DA@ , the sound speed into the gas is equal to:(9)
c i n 2 = γ R M W i n T i n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGJbWaaS baaSqaaiaadMgacaWGUbaabeaakmaaCaaaleqabaGaaGOmaaaakiab g2da9iabeo7aNnaalaaabaGaamOuaaqaaiaad2eacaWGxbWaaSbaaS qaaiaadMgacaWGUbaabeaaaaGccaWGubWaaSbaaSqaaiaadMgacaWG Ubaabeaaaaa@448F@
and if V i n > c i n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGwbWaaS baaSqaaiaadMgacaWGUbaabeaakiabg6da+iaadogadaWgaaWcbaGa amyAaiaad6gaaeqaaaaa@3D4D@ , then V i n = c i n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGwbWaaS baaSqaaiaadMgacaWGUbaabeaakiabg2da9iaadogadaWgaaWcbaGa amyAaiaad6gaaeqaaaaa@3D4B@ the velocity is major by the sound speed velocity.
  1. If the injector temperature is known:

    d m d t = v γ ( γ 1 ) C V T i n d P d t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaai aadsgacaWGTbaabaGaamizaiaadshaaaGaeyypa0ZaaSaaaeaacaWG 2baabaGaeq4SdC2aaeWaaeaacqaHZoWzcqGHsislcaaIXaaacaGLOa GaayzkaaGaam4qamaaBaaaleaacaWGwbaabeaakiaadsfadaWgaaWc baGaamyAaiaad6gaaeqaaaaakmaalaaabaGaamizaiaadcfaaeaaca WGKbGaamiDaaaaaaa@4B34@

  2. If the temperature in the tank is known:

    d m d t = V γ ( γ 1 ) C V T 2 ( T d P d t P d T d t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaai aadsgacaWGTbaabaGaamizaiaadshaaaGaeyypa0ZaaSaaaeaacaWG wbaabaGaeq4SdC2aaeWaaeaacqaHZoWzcqGHsislcaaIXaaacaGLOa GaayzkaaGaam4qamaaBaaaleaacaWGwbaabeaakiaadsfadaahaaWc beqaaiaaikdaaaaaaOWaaeWaaeaacaWGubWaaSaaaeaacaWGKbGaam iuaaqaaiaadsgacaWG0baaaiabgkHiTiaadcfadaWcaaqaaiaadsga caWGubaabaGaamizaiaadshaaaaacaGLOaGaayzkaaaaaa@51C8@ or m = P V ( γ 1 ) C V T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGTbGaey ypa0ZaaSaaaeaacaWGqbGaamOvaaqaamaabmaabaGaeq4SdCMaeyOe I0IaaGymaaGaayjkaiaawMcaaiaadoeadaWgaaWcbaGaamOvaaqaba GccaWGubaaaaaa@41A0@

    T i n = T 2 γ ( d P d t T d P d t P d T d t ) or T i n = T γ + m γ d T d t d m d t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGubWaaS baaSqaaiaadMgacaWGUbaabeaakiabg2da9maalaaabaGaamivamaa CaaaleqabaGaaGOmaaaaaOqaaiabeo7aNbaadaqadaqaamaalaaaba WaaSaaaeaacaWGKbGaamiuaaqaaiaadsgacaWG0baaaaqaaiaadsfa daWcaaqaaiaadsgacaWGqbaabaGaamizaiaadshaaaGaeyOeI0Iaam iuamaalaaabaGaamizaiaadsfaaeaacaWGKbGaamiDaaaaaaaacaGL OaGaayzkaaGaaGzbVlaab+gacaqGYbGaaGzbVlaadsfadaWgaaWcba GaamyAaiaad6gaaeqaaOGaeyypa0ZaaSaaaeaacaWGubaabaGaeq4S dCgaaiabgUcaRmaalaaabaGaamyBaaqaaiabeo7aNbaadaWcaaqaam aalaaabaGaamizaiaadsfaaeaacaWGKbGaamiDaaaaaeaadaWcaaqa aiaadsgacaWGTbaabaGaamizaiaadshaaaaaaaaa@63C4@

  3. If the temperature in the tank is constant:

    T i n = T γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGubWaaS baaSqaaiaadMgacaWGUbaabeaakiabg2da9maalaaabaGaamivaaqa aiabeo7aNbaaaaa@3CE4@