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The Impulse Response of an Exponential Assembly; Reponse d'un Assemblage Exponentiel aux Impulsions; Impul'snaya kharakteristika ehksponentsial'noj sborki; Respuesta de un Conjunto Exponencial a los Impulsos

## Abstract

A spatial-dependent transfer function of an exponential assembly with rectangular geometry and the neutron source located at the origin in the middle of one side has been derived using diffusion theory and the usual approximations. If the end correction factor is neglected, the result G (x, y, z, s) = ({Delta}#Greek Phi Symbol#(x, y, z, s))/{Delta}S (o, o, o, o, s) = 2/abD {sup {infinity}}{Sigma}{sub m} {sub =} {sub 1} {sup {infinity}}{Sigma}{sub n} {sub =} {sub 1} 1/({rho}{sub mn}(s) cos (m{pi}x)/a cos (n{pi}y)/b e{sup -{rho}}mn{sup (s)Z} (1) where {rho}{sub mn}{sup (s)} = [s/(VD) + B{sup 2}{sub Up-Tack mn} - B{sup 2}{sub m}]{sup 1/2} = [s/VD - {gamma}{sup 2}{sub mn}]{sup 1/2} (2) The inverse Laplace transform of Eq. (1) gives the spatial dependent impulse response function (Green's function) to be g (x, y, z, t) = 2/ab {radical}v/{pi}D 1/{radical}t e{sup -z{sup 2/4}} {sup VDt} {sup {infinity}}{Sigma}{sub m} {sub =} {sub 1} {sup {infinity}}{Sigma}{sub n} {sub =} {sub 1} cos m{pi}x/a cos n{pi}y/b e{sup -VD{gamma}{sup 2mn{sup t}}} (3) If only the terms of Eq. (3) outside the double summation are considered (i.e. multiplicative and leakage effects are neglected) the result is 2/ab {radical}V/{pi}D 1/{radical}t e{sup -z{sup 2/4}} {sup VDt} = 4V/ab P(z),  More>>
Authors:
Uhrig, R. E. [1]
1. University of Florida, Gainesville, FL (United States)
Publication Date:
Apr 15, 1964
Product Type:
Conference
Report Number:
IAEA-SM-42/43
Resource Relation:
Conference: Symposium on Exponential and Critical Experiments, Amsterdam (Netherlands), 2-6 Sep 1963; Other Information: 45 refs., 18 figs.; Related Information: In: Exponential and Critical Experiments. Vol. III. Proceedings of the Symposium on Exponential and Critical Experiments| 494 p.
Subject:
21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS; APPROXIMATIONS; GAUSS FUNCTION; GREEN FUNCTION; HARMONICS; HEAVY WATER; HEAVY WATER MODERATED REACTORS; LAPLACE TRANSFORMATION; MODERATORS; NATURAL URANIUM; NEUTRON DENSITY; NEUTRON SOURCES; NUCLEAR FUELS; PULSES; REACTOR PHYSICS; RESPONSE FUNCTIONS; THERMAL NEUTRONS; TRANSFER FUNCTIONS
OSTI ID:
22123056
Research Organizations:
International Atomic Energy Agency, Vienna (Austria)
Country of Origin:
IAEA
Language:
English
Other Identifying Numbers:
Other: ISSN 0074-1884; TRN: XA13M2712078858
Submitting Site:
INIS
Size:
page(s) 205-239
Announcement Date:
Aug 30, 2013

## Citation Formats

Uhrig, R. E. The Impulse Response of an Exponential Assembly; Reponse d'un Assemblage Exponentiel aux Impulsions; Impul'snaya kharakteristika ehksponentsial'noj sborki; Respuesta de un Conjunto Exponencial a los Impulsos. IAEA: N. p., 1964. Web.
Uhrig, R. E. The Impulse Response of an Exponential Assembly; Reponse d'un Assemblage Exponentiel aux Impulsions; Impul'snaya kharakteristika ehksponentsial'noj sborki; Respuesta de un Conjunto Exponencial a los Impulsos. IAEA.
Uhrig, R. E. 1964. "The Impulse Response of an Exponential Assembly; Reponse d'un Assemblage Exponentiel aux Impulsions; Impul'snaya kharakteristika ehksponentsial'noj sborki; Respuesta de un Conjunto Exponencial a los Impulsos." IAEA.
@misc{etde_22123056,
title = {The Impulse Response of an Exponential Assembly; Reponse d'un Assemblage Exponentiel aux Impulsions; Impul'snaya kharakteristika ehksponentsial'noj sborki; Respuesta de un Conjunto Exponencial a los Impulsos}
author = {Uhrig, R. E.}
abstractNote = {A spatial-dependent transfer function of an exponential assembly with rectangular geometry and the neutron source located at the origin in the middle of one side has been derived using diffusion theory and the usual approximations. If the end correction factor is neglected, the result G (x, y, z, s) = ({Delta}#Greek Phi Symbol#(x, y, z, s))/{Delta}S (o, o, o, o, s) = 2/abD {sup {infinity}}{Sigma}{sub m} {sub =} {sub 1} {sup {infinity}}{Sigma}{sub n} {sub =} {sub 1} 1/({rho}{sub mn}(s) cos (m{pi}x)/a cos (n{pi}y)/b e{sup -{rho}}mn{sup (s)Z} (1) where {rho}{sub mn}{sup (s)} = [s/(VD) + B{sup 2}{sub Up-Tack mn} - B{sup 2}{sub m}]{sup 1/2} = [s/VD - {gamma}{sup 2}{sub mn}]{sup 1/2} (2) The inverse Laplace transform of Eq. (1) gives the spatial dependent impulse response function (Green's function) to be g (x, y, z, t) = 2/ab {radical}v/{pi}D 1/{radical}t e{sup -z{sup 2/4}} {sup VDt} {sup {infinity}}{Sigma}{sub m} {sub =} {sub 1} {sup {infinity}}{Sigma}{sub n} {sub =} {sub 1} cos m{pi}x/a cos n{pi}y/b e{sup -VD{gamma}{sup 2mn{sup t}}} (3) If only the terms of Eq. (3) outside the double summation are considered (i.e. multiplicative and leakage effects are neglected) the result is 2/ab {radical}V/{pi}D 1/{radical}t e{sup -z{sup 2/4}} {sup VDt} = 4V/ab P(z), (4) where P(z) is a Gaussian distribution term of the form P(z) = 1/o{radical}2{pi} e{sup -z{sup 2/2{gamma}{sup 2}}} (5) with a time dependent standard deviation o = {radical}2 VDt. (6) Hence, a pulse of thermal neutrons introduced at the origin spreads Out with time in a symmetrical manner about the z = 0 plane. However, the variation of amplitude with time at any position along the z axis shows that a peak value of neutron density does move out from the origin with a decreasing amplitude. Although the multiplication and transverse leakage influence the characteristics of the disturbance, it does propagate away from the source in a manner similar to the propagation of neutron waves. The propagation of a thermal-neutron pulse has been demonstrated experimentally in 1962 at the University of Florida using a pulsed-neutron source and a ''Thermalizer box''. However, the method used for the experiments reported here was the cross correlation between the pseudo-random binary (off-on) variation of source strength and the resulting variation of neutron density in the exponential assembly. Data are given for experiments carried out on both light- and heavy-water moderated assemblies using natural uranium. The results are discussed in terms of the theoretical relations derived and the physical phenomena taking place. The validity of the derived relationships and the need for considering higher harmonics for various arrangements of fuel and moderator are discussed briefly. (author) [French] En utilisant la theorie de la diffusion et les approximations habituelles, l'auteur a etabli une fonction de transfert dependant de l'espace d'un assemblage exponentiel a geometrie rectangulaire, l'origine etant au milieu de l'un des cotes ou la source de neutrons est placee. Si Ton neglige le facteur de correction final, on obtient l'expression suivante: G (x, y, z, s) = ({Delta}#Greek Phi Symbol#(x, y, z, s))/{Delta}S (o, o, o, o, s) = 2/abD {sup {infinity}}{Sigma}{sub m} {sub =} {sub 1} {sup {infinity}}{Sigma}{sub n} {sub =} {sub 1} 1/({rho}{sub mn}(s) cos (m{pi}x)/a cos (n{pi}y)/b e{sup -{rho}}mn{sup (s)Z} (1) dans laquelle {rho}{sub mn}{sup (s)} = [s/(VD) + B{sup 2}{sub Up-Tack mn} - B{sup 2}{sub m}]{sup 1/2} = [s/VD - {gamma}{sup 2}{sub mn}]{sup 1/2} (2) L'inverse de la transformee de Laplace de l'equation (1) donne comme fonction de reponse aux impulsions dependant de l'espace (fonction de Green) g (x, y, z, t) = 2/ab Square-Root v/{pi}D 1/ Square-Root t e{sup -z{sup 2/4}} {sup VDt} {sup {infinity}}{Sigma}{sub m} {sub =} {sub 1} {sup {infinity}}{Sigma}{sub n} {sub =} {sub 1} cos m{pi}x/a cos n{pi}y/b e{sup -VD{gamma}{sup 2mn{sup t}}} (3) Si l'on considere uniquement les termes de l'equation (3) sans tenir compte de la double sommation (c'est-a-dire si l'on neglige les effets de multiplication et des effets de fuite), on obtient le resultat suivant: 2/ab Square-Root V/{pi}D 1/ Square-Root t e{sup -z{sup 2/4}} {sup VDt} = 4V/ab P(z), (4) dans lequel P(z) est un terme de distribution normale de la forme P(z) = 1/o Square-Root 2{pi} e{sup -z{sup 2/2{gamma}{sup 2}}} (5) avec un ecart type qui est fonction du temps o = Square-Root 2 VDt. (6) Par consequent, une impulsion de neutrons thermiques fournie a l'origine se deploie avec le temps de facon symetrique selon le plan z = 0. Toutefois, la variation d'amplitude en fonction du temps en un point quelconque le long de l'axe de z montre qu'une valeur maximum du nombre volumique de neutrons se deplace bien a partir de l'origine avec une amplitude decroissante. Bien que la multiplication et la fuite transversale aient un effet sur les caracteristiques de la perturbation, cette perturbation se propage bien a partir de la source de la meme facon que les ondes neutroniques. On a fait une experience sur la propagation d'une impulsion de neutrons thermiques en 1962 a l'Universite de Floride en utilisant une source de neutrons puises et une 'boftede thermalisation'. Toutefois, l'auteur a utilise pour les experiences ci-dessus mentionnees la methode de correlation entre la variation binaire pseudoaleatoire (ouvert-ferme) de l'intensite de la source et la variation du nombre volumique de neutrons qui en resulte dans l'assemblage exponentiel. L'auteur fournit des donnees sur les experiences faites avec des assemblages a uranium naturel et a eau ordinaire et des assemblages a uranium naturel et a eau lourde. U examine les resultats, compte tenu des relations theoriques qu'il a etablies et des phenomenes physiques qui se produisent. Il examine brievement la validite de ces relations et la necessite de tenir compte d'harmoniques d'ordre superieur pour les diverses dispositions du combustible et du ralentisseur. (author) [Spanish] El autor recurre a la teoria de la difusion y a las aproximaciones corrientes a fin de derivar una funcion de transferencia dependiente del espacio para un conjunto exponencial con geometria rectangular y cuya fuente neutronica esta situada en el centro de uno de los lados. Si se desprecia el factor de correccion final, el resultado es: G (x, y, z, s) = ({Delta}#Greek Phi Symbol#(x, y, z, s))/{Delta}S (o, o, o, o, s) = 2/abD {sup {infinity}}{Sigma}{sub m} {sub =} {sub 1} {sup {infinity}}{Sigma}{sub n} {sub =} {sub 1} 1/({rho}{sub mn}(s) cos (m{pi}x)/a cos (n{pi}y)/b e{sup -{rho}}mn{sup (s)Z} (1) donde {rho}{sub mn}{sup (s)} = [s/(VD) + B{sup 2}{sub Up-Tack mn} - B{sup 2}{sub m}]{sup 1/2} = [s/VD - {gamma}{sup 2}{sub mn}]{sup 1/2} (2) La transformacion laplaciana inversa de la ecuacion (1) da para la funcion de respuesta al impulso, dependiente del espacio (funcion de Green), g (x, y, z, t) = 2/ab Square-Root v/{pi}D 1/ Square-Root t e{sup -z{sup 2/4}} {sup VDt} {sup {infinity}}{Sigma}{sub m} {sub =} {sub 1} {sup {infinity}}{Sigma}{sub n} {sub =} {sub 1} cos m{pi}x/a cos n{pi}y/b e{sup -VD{gamma}{sup 2mn{sup t}}} (3) Si en la ecuacion (3) se desprecia la doble suma (es decir, los efectos multiplicativos y de escape) el resultado sera 2/ab Square-Root V/{pi}D 1/ Square-Root t e{sup -z{sup 2/4}} {sup VDt} = 4V/ab P(z), (4) donde P(z) constituye un termino de distribucion gaussiana de la forma P(z) = 1/o Square-Root 2{pi} e{sup -z{sup 2/2{gamma}{sup 2}}} (5) con una desviacion standard, dependiente del tiempo, o = Square-Root 2 VDt. (6) Por tanto, un impulso de neutrones termicos introducido en el origen se distribuye con el tiempo simetricamente por el plano z = O. No obstante, la variacion de la amplitud con el tiempo en cualquier posicion a lo largo del eje z muestra que, partiendo del origen, un maximo de la densidad neutronica se desplaza con amplitud decreciente. Aunque la multiplicacion y las perdidas transversales ejercen influencia sobre las caracteristicas de la perturbacion, esta se propaga alejandose de la fuente de modo similar a la propagacion de ondas neutronicas. En 1962, se demostro experimentalmente en la Universidad de Florida la propagacion de un impulso de neutrones termicos empleando una fuente pulsante y una 'caja termalizadora'. Sin embargo, el metodo empleado para los experimentos descritos en la presente memoria consistio en establecer una correlacion entre la variacion binaria (parormarcha) pseudo-aleatoria de la intensidad de la fuente y las variaciones de la densidad neutronica a que da lugar en el conjunto exponencial. La memoria expone datos relativos a experimentos llevados a cabo en conjuntos de uranio natural moderados con agua ligera y agua pesada. Discute los resultados basandose en las relaciones teoricas establecidas y en los fenomenos fisicos correspondientes. Examina sucintamente la validez de las relaciones deducidas y la necesidad de tener en cuenta armonicos superiores en el caso de diversas disposiciones de combustible y moderador. (author) [Russian] Vyvoditsja prostranstvenno zavisimaja funkcija perenosa jeksponencial'noj sborki s prjamougol'noj geometriej i nejtronnym istochnikom, raspolozhennym v ishodnom punkte v seredine odnoj storony . Pri jetom ispol'zujutsja teorija diffuzii i obychnyh priblizhenij. Esli ne uchityvat' konechnyj kojefficient popravki, to poluchaetsja sledujushhij rezul'tat: G (x, y, z, s) = ({Delta}#Greek Phi Symbol#(x, y, z, s))/{Delta}S (o, o, o, o, s) = 2/abD {sup {infinity}}{Sigma}{sub m} {sub =} {sub 1} {sup {infinity}}{Sigma}{sub n} {sub =} {sub 1} 1/({rho}{sub mn}(s) cos (m{pi}x)/a cos (n{pi}y)/b e{sup -{rho}}mn{sup (s)Z} (1) {rho}{sub mn}{sup (s)} = [s/(VD) + B{sup 2}{sub Up-Tack mn} - B{sup 2}{sub m}]{sup 1/2} = [s/VD - {gamma}{sup 2}{sub mn}]{sup 1/2} (2) Obratnoe laplasovo preobrazovanie uravnenija (A -1) daet prostranstvenno zavisimuju funkciju impul'snoj harakteristiki (funkcija Grina) sledujushhego vida: g (x, y, z, t) = 2/ab Square-Root v/{pi}D 1/ Square-Root t e{sup -z{sup 2/4}} {sup VDt} {sup {infinity}}{Sigma}{sub m} {sub =} {sub 1} {sup {infinity}}{Sigma}{sub n} {sub =} {sub 1} cos m{pi}x/a cos n{pi}y/b e{sup -VD{gamma}{sup 2mn{sup t}}} (3) Esli rassmatrivajutsja tol'ko chleny uravnenija (A-3) vne dvojnogo slozhenija (t.e. ne uchityvajutsja faktory razmnozhenija i utechki)* to poluchaetsja sledujushhij rezul'tat: 2/ab Square-Root V/{pi}D 1/ Square-Root t e{sup -z{sup 2/4}} {sup VDt} = 4V/ab P(z), (4) gde P (z) javljaetsja gaussovym chlenom raspredelenija formy: P(z) = 1/o Square-Root 2{pi} e{sup -z{sup 2/2{gamma}{sup 2}}} (5) pri zavisimom ot vremeni standartnom otklonenii o = Square-Root 2 VDt. (6) Takim obrazom, impul's teplovyh nejtronov, vvedennyj v ishodnoj punkt, rasprostranjaetsja so vremenem simmetricheski po ploskosti z =o. Odnako izmenenie amplitudy so vremenem v ljubom meste vdol' z pokazyvaet, chto maksimal'naja velichina plotnosti nejtronov ishodit iz ishodnogo punkta s umen'sheniem amplitudy. Hotja razmnozhenie i obratnaja utechka vlijajut na harakteristiki vozmushhenija, jeto rasprostranjaetsja ot istochnika podobno nejtronnym volnam. Rasprostranenie impul'sa teplovyh nejtronov bylo prodemonstrirovano jeksperimental'no v 1962 godu vo Floridskom universitete pri ispol'zovanii impul'snogo istochnika nejtronov i {sup j}ashhika termalizatora{sup .} Na jetot raz byl ispol'zovan metod perekrestnoj korreljacii psevdo-besporjadochnogo dvojnogo (vykljuchenija-vkljuchenija) izmenenija sily istochnika i izmenenija v rezul'tate jetogo plotnosti nejtronov v jeksponencial'noj sborke. Privodjatsja dannye dlja jeksperimentov, provedennyh na sborkah s zamedleniem kak na legkoj, tak i na tjazheloj vode, na kotoryh ispol'zuetsja prirodnyj uran. Obsuzhdajutsja rezul'taty s tochki zrenija poluchennoj teoreticheskoj svjazi i fizicheskih javlenij. Kratko obsuzhdajutsja dejstvitel'nost' ustanovlennyh svjazej i neobhodimost' rassmotrenija bolee vysokih garmonikov dlja razlichnyh raspolozhenij gorjuchego i zamedlitelja. (author)}
place = {IAEA}
year = {1964}
month = {Apr}
}