j465 Appendix B Production Equations in Health Physics

j465 Appendix B Production Equations in Health Physics B.1 Introduction The assumption that radioactive material enters a system at a constant rate leads to a set of production equations that describe a broad class of phenomena encountered by health physicists. Equations governing activation, buildup of radioactive material on a filter or demineralizer, deposition of material on a surface from a radioactive plume, and release of material into a room are examples of phenomena described consistently by production equations. This appendix describes production equations and their applications in a wide variety of health physics areas. B.2 Theory In health physics applications, the rate of change of radioactive material in a system is described by first-order linear differential equations that have exponential solutions. Since exponential forms appear throughout the field, it is not unexpected that phenomena describing the accumulation of radioactive material have a similar mathematical structure. This text refers to these structures as production equations. To formulate a general form of production relationship, consider the time rate of change of activity _A associated with the continuous introduction of a radionuclide into a system or structure. For a given radionuclide _A ¼ Pe Kt ; ðb:1þ where P is the production term or the rate at which activity is added to the system (e.g., room, accelerator target, or filter paper), K is the total removal rate of the radionuclide from the system, and t is the time from the start of production. To simplify the equation resulting from the integration of Equation B.1, P is assumed to be constant. The production term has units of activity per unit time (Bq/s). Examples of the production term for a variety of physical phenomena are provided in Table B.1. Health Physics in the 21st Century. Joseph John Bevelacqua Copyright Ó 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40822-1

466j Appendix B Table B.1 Examples of production terms in health physics applications. Physical phenomena P (Bq/s) Definition of terms (units) Activation of material in an accelerator Activation of material in a reactor Deposition of radioactive material in a demineralizer bed Deposition of radioactive material in a filter Nsjl Nsjl CFe CFe N ¼ number of target atoms of the nuclide being activated (atoms) s ¼ activation cross section for the specific activation reaction (b/atom or cm 2 /atom) j ¼ activating flux of a beam of particles (particles/cm 2 s) l ¼ radioactive disintegration constant (s 1 ) N ¼ number of target atoms of the nuclide being activated (atoms) s ¼ activation cross section for the specific activation reaction (b/atom or cm 2 /atom) j ¼ activating flux of neutrons (neutrons/ cm 2 s) l ¼ radioactive disintegration constant (s 1 ) C ¼ influent activity concentration of an isotope entering the demineralizer (Bq/m 3 ) F ¼flow rate of fluid through the demineralizer (m 3 /s) e ¼ isotope specific removal efficiency of the demineralizer bed C ¼ influent activity concentration of an isotope entering the filter (Bq/m 3 ) F ¼ flow rate of fluid through the filter (m 3 /s) e ¼ isotope specific removal efficiency of the filter Surface deposition from a radioactive plume ws w ¼ ground deposition rate (Bq/m 2 s) S ¼ surface area of the deposition (m 2 ) Inhalation of radioactive material Surface deposition from a leaking radioactive fluid Airborne entry of 222 Rn into a home Cr CF CF C ¼ air concentration of radioactive material (Bq/m 3 ) r ¼ breathing rate (m 3 /s) C ¼ activity concentration of the isotope in the fluid leaking onto the surface (Bq/m 3 ) F ¼ leak rate of the fluid onto the surface (m 3 /s) C ¼ air concentration of 222 Rn entering the home (Bq/m 3 ) F ¼air infiltration rate entering the home (m 3 /s)

B.2 Theory j467 Table B.1 (Continued) Physical phenomena P (Bq/s) Definition of terms (units) Release of radioactive material from a stack Release of radioactive material into a room CF Q C ¼ air concentration of radioactive material being released (Bq/m 3 ) from a stack F ¼stack flow rate (m 3 /s) Q¼ release rate of airborne radioactive material into the room (Bq/s) When using Equation B.1, it is important that the production equation be applied separately for each radionuclide of interest. The quantities P and K depend on the radionuclide half-life as well as on its physical and chemical properties. The total removal rate has numerous components. The most common components are derived from radioactive decay (l), biological decay (l b ), or ventilation (l v ). Explicit forms for these removal rates are l ¼ lnð2þ=t 1=2 ; ðb:2þ l b ¼ lnð2þ=t b 1=2 ; l v ¼ F=V; ðb:3þ ðb:4þ where T 1/2 is the physical half-life, T1=2 b is the biological half-life, F is the ventilation flow rate of the system, and V is the free air volume of the system. The total removal rate K ¼ lþl b þl v þ ðb:5þ is the sum of the individual removal rates as they apply to the problem of interest. Not all terms in Equation B.5 appear in each application. The specific application of removal rates is addressed in subsequent discussion. Equation B.1 can be integrated with respect to time from t ¼ 0 to t ¼ T where the time T is the end of the production interval: ð T 0 _A dt ¼ AðTÞ ¼ ð T 0 Pe Kt dt ¼ P ð T 0 e Kt dt: ðb:6þ In Equation B.6, we assume that no activity is initially present in the system (A(0) ¼ 0). Using this condition leads to the result AðTÞ ¼ P K ð1 e KT Þ: ðb:7þ Equation B.7 provides a relationship describing the buildup of activity during the time that the production term is active. For KT 1, the system activity reaches its

468j Appendix B maximum value. Accordingly, Equation B.7 is written as Að1Þ ¼ A eq ¼ P K : ðb:8þ The saturation or equilibrium activity is the maximum activity that can be achieved in the system. If T is defined as the time during which the production term is active and t describes the time after the production ceases, Equation B.7 is rewritten to describe the activity variation following the production interval and during the subsequent decay period: AðtÞ ¼ P K ð1 e KT Þe kt ðb:9þ where k is the total removal rate postproduction, that is, during the decay time t.asa matter of specificity, t ¼ 0 corresponds to the time when production ceases. B.3 Examples A number of examples are provided to illustrate the utility of the general production equation. These examples include (1) the activation of a target by an accelerator beam or reactor neutron source, (2) buildup of activity on a filter or demineralizer, (3) buildup of activity in a pond, and (4) release of activity into a room. B.3.1 Activation Activation is a process described by the reaction C(c, d)d, during which the radiation of type c strikes a target nucleus C and produces a radioactive nucleus D and radiation of type d. Examples of activation reactions include 59 Coðn; gþ 60 Co, 16 Oðn; pþ 16 N, 27 Alðn; aþ 24 Na, and 3 Hðp; nþ 3 He. Using the generalized production equation (Equation B.9) and the production term from Table B.1 leads to a relationship that describes the activity in the target as a function of time: A ¼ Nsj½1 e lt Še lt ; ðb:10þ where N, s, and j are defined in Table B.1. For nongaseous products, the removal rates (K and k) are equal to the physical decay constant (l). T is the irradiation time, that is, the time the target is irradiated by the accelerators beam or the time the material to be activated is exposed to the reactors neutron fluence rate (flux). The time after the reactor is shut down or the accelerator beam is terminated is t. The steady-state (saturation) or equilibrium activity is Nsj. The application of Equation B.10 is further illustrated by considering the activation of 59 Co by thermal neutrons. In this example, N is the number of 59 Co atoms in the

B.3 Examples j469 target, s is the microscopic cross section for the 59 Coðn thermal ; gþ 60 Co reaction, j is the number of thermal neutrons per cm 2 s, and l is the 60 Co decay constant. Equation B.10 is applied separately for each activated species. B.3.2 Demineralizer Activity Ion exchange is a process used in a variety of nuclear facilities to reduce the radioactive ion content of water by removing radioactive ions and replacing them with nonradioactive ions. The device in which the ion exchange occurs is commonly called a demineralizer. The activity that accumulates within a demineralizer bed is also obtained from Equation B.9 and Table B.1: A ¼ CFe l ½1 e lt Še lt : ðb:11þ Equation B.11 is also to be applied individually for each isotope trapped in the demineralizer bed. In Equation B.11, C, F, and e are defined in Table B.1, l is the physical decay constant of the trapped material, T is the time the demineralizer in online (valved in) and removing radioactivity from the influent stream, and t is the time after the demineralizer is no longer in service (valved out). For the demineralizer application, the total removal rate is just the physical decay constant. Equation B.11 also applies to filters. The saturation activity for both filters and demineralizers is CFe/l. B.3.3 Surface Deposition The deposition of radioactive material onto a surface from an airborne plume is also described by a production equation. Again, using Table B.1 and Equation B.9, the activity deposited onto a surface is A ¼ ws K ½1 e KT Še kt ; ðb:12þ and ws is defined in Table B.1. The removal rates k and K are discussed below. Equation B.12 is used to illustrate the versatility of the production equation. Assuming that there is a continuous release of radioactive material from a plume and that an equilibrium has been reached, an expression for the equilibrium activity that has been removed from the plume and deposited on a surface of area S is written as A eq ¼ ws K : ðb:13þ

470j Appendix B If it is also assumed that the material deposits on the surface of a stationary body of water, such as a pond, then Equation B.13 still applies and K ¼ lþl b ; ðb:14þ where l b is the biological removal rate from the pond. If the radionuclide deposited onto the surface of the pond is also soluble in the pond water, and instantaneous mixing of the radionuclide within the pond occurs, then the equilibrium concentration C eq of the radionuclide in the pond water is determined from the relation C eq ¼ A eq V ; ðb:15þ where V is the volume of water in the pond. Determine the equilibrium concentration of a radionuclide in a pond using Equations B.13 and B.15: C eq ¼ ws KV : ðb:16þ The production concept can also be extended to calculate the equilibrium concentration in an organism, such as a fish, living in the pond. The equilibrium activity concentration per unit mass (Bq/kg) in the fish (C eq-fish ) is written as C eq-fish ¼ IC eq K 0 ; ðb:17þ where I is the intake of pond water by the fish (m 3 /kg(fish) s) and K 0 is the total removal rate of the isotope from the fish K 0 ¼ lþl 0 b ; ðb:18þ where l 0 b is the biological removal rate from the fish. A careful examination of Equation B.18 indicates that the term C eq I is just P per unit mass of the fish. Equation B.18 is another application of the production equation, Equation B.9. B.3.4 Release of Radioactive Material into a Room The release of airborne radioactive material into a room is obtained from Equation B.9 and Table B.1: A ¼ Q K ½1 e KT Še kt ; ðb:19þ where the removal of radioactive material includes both physical decay and ventilation terms

References j471 K ¼ k ¼ lþ F V : ðb:20þ In Equation B.20, the ventilation rate is assumed to be constant during the production and postproduction periods. B.4 Conclusions The use of production equations has been shown to provide a unified explanation for a wide variety of phenomena encountered in health physics. The specific application determines the P, K, and k values, but the form of the equation remains the same. The use of production equations greatly simplifies the understanding of a variety of health physics concepts that appear to involve dissimilar phenomena. References Bevelacqua, J.J. (1995) Contemporary Health Physics: Problems and Solutions, John Wiley & Sons, Inc., New York. Bevelacqua, J.J. (1999) Basic Health Physics: Problems and Solutions, John Wiley & Sons, Inc., New York. Bevelacqua, J.J. (2003) Production Equations in Health Physics. Radiation Protection Management, 20 (6), 9. Cember, H. (1996) Introduction to Health Physics, 3rd edn, McGraw-Hill, New York. Turner, J.E. (1995) Atoms, Radiation, and Radiation Protection, 2nd edn., John Wiley & Sons, Inc., New York.