DE3912995A1 - Metal fission prod. deposit determn. for high temp. reactor - measures activity distributions occurring on surfaces of metals and alloys - Google Patents

Abstract

Analysis method for reactor experiments measures fission prodt. depositions, independently of additional time consuming laboratory experiments with the measuring of the material parameters. From the analysis of the reactor experiments forecasts can be made wrt activities at metallic components for determining opt faults and correlated fault times, to facilitate service of high temp. reactors. USE - Forecast and determn. of fission prod depositions at metallic components. Used for cooling gas temp. of 400 - 900 deg.C. Assists in maintenance of components and their decontamination.

Description

1.1 Technical Applications

The named method is for the investigation of the fission product deposits and their quantitative determination of occurring activity distributions on metal surfaces, for reactor components in the primary circuit, in HTR systems been developed.

The method is for a cooling gas temperature range of about 400 to 900 ° C in HTR systems applicable.

From the reactor experiments should by means of the method the necessary parameters for the determination of the fission product deposits, from measured activity distributions on the Metal surfaces in the experimental arrangements to be determined. A prediction of occurring activity distributions until to a possible incident appears only possible be when the parameters for the adsorption and desorption processes from reactor experiments at different Measuring times, at about the same hot gas activities and with Materials for the reactor components, at about the same Corrosion levels and compositions of the metal alloy have been set.

The method is based on a modified model for Description of physisorption processes and is therefore regardless of reactor safety in other areas as well applicable when much of the chemical compounds in the temperature ranges to be considered thermodynamically is unstable.

The focus of technical application, in the field of Reactor safety for this process include operation of HTR systems, the accident analyzes and the maintenance of components, the deposition of released fission products are exposed in the primary circuit and their Decontamination [1]. The procedure is not applicable for the cleavage product deposits and diffusion processes in graphite.  

1.2 Purpose of the procedure

The purpose of the procedure arises from the possible Fault analysis to release the fission products the primary circuit. Two major types of accident are the ingress of water in the Primary circuit through leaking steam generators (wash-off processes) and the pressure relief in the cooling gas circuit in the release of radioactive dusts from material surfaces (lift-off operations) by shear forces.

Water penetration is the concentration dissolved by the water of cleavage products of importance from the steam generator surface is replaced by water and water vapor. For qantitative recording of the solution process is the Starting concentration, which can be calculated from the predicted, local and temperature-dependent activity distribution with help determine this method.

With this procedure is a calculation of the fission product inventory on the metal surfaces of the reactor components possible, so that be determined by experiments previously can be compared to what proportion of radioactive dusts released to the determined inventory at pressure reliefs is, even if with the method of chemically to dust bound fraction of fission products not detected can be.

From the reactor experiments, to measure the course of activity on wall materials (pile and out of pile experiments), the cleavage product retention capacity can have good thermal conductivity Metal alloys can be studied using this method. These examinations are then for the maintenance of components and their decontamination of importance.  

2. Previous state of the physical models Description of fission product deposits

To give a rough overview of the current state of the art physical description of the cleavage product deposits too give, here are two models, by Neill and Kress and by Iniotakis [3]. These two models serve as starting models for one Modified model, as a physical basis, for Determination of fission product deposits with the help of the still to be described.

2.1 Model of Neill and Kress

For all the models to be considered, basically two balance equations are considered, once for the flow and convection process and for the balance of the fission products on the material surface. If the mass transfer is considered in the flow and convection process, then the following balance equation can be established for a grade i of fission product nuclides for the density change in the cooling gas flow

N _i  is the seal for the speciesi of a cleavage product in kg / m³.
N _wi  is the density for the varietyi a cleavage product on the Surface of the material wall in kg / m³.
A is the cross-sectional area, z. B for a pipe through the the cooling gas flow passes through in an experimental setup, in m².
P is the circumference for the cross-sectional area in m.
 is the mean flow velocity of the cooling gas flow in m / s.
H is the mass transfer coefficient in m / s.
λ _i  is the radioactive decay constant in 1 / s.
x is the location in the direction of flow at which the density change is considered, in m.

The following Figure 1, from the literature [2], is intended to individual procedures for establishing the balance sheet equation (2.1.1) clarify again.

For the individual mass flows, the following terms are obtained:

incoming mass flow  =N _i  ·  ·A.
exiting mass flow

Mass flow during mass transfer to the material wall _St  =H ·(N _i  -N _wi · ·P · Dx,
The temporal change of the fission products is

The decrease of mass flow between incoming and exiting mass flow is

The decrease in the mass flow through radioactive decay is λ _i · A · N _i · d x.

The mass balance M _i , for the sort i of the fission product nuclides on the material surface, is calculated according to [2] by the following balance equation

shown.

To determine the density N _wi , the fission products on the wall surface, the Langmuir adsorption isotherm is used. The equilibrium assumed here is an exchange of the particles on the surface with the fission product nuclides in the gas phase in the cooling gas stream.

Since you later the equations for the material flow densities at the adsorption and desorption needed again the equations for the derivation of the Langmuir adsorption isotherm already anticipated. If it is assumed that the material flow densities for Adsorption and desorption or adsorption and Desorption rates at constant temperature are equal to each other, then the equation for the Langmuir adsorption isotherm for the calculation of Derive fission product density on the material surface.  

For the material flow density, in the adsorption or adsorption, can one use the equation according to literature [2]?

use.

If the particles or the cleavage products are adsorbed on the material surface at a point during mass transfer, f (R) as the surface fraction is the probability that the molecules have a vacancy on the surface during adsorption. The coverage-dependent function f (R) can then be expressed

f (R) = 1 - R (2.1.6) [4]

to be written.

The individual quantities in the equations (2.1.3) to (2.1.6) have the following names:

r _a = d n _a / d t is the mass flow density or velocity for the adsorption of the particles in atoms / m².s,
s is the sticking probalility,
Z is the number of impacts, for the number of particles hitting the surface, in atoms / m² · s,
p is the gas pressure (partial pressure of the nuclides in the cooling gas flow) in N / m² or Pa,
m ₀ is the mass of the impinging particle or nuclide in kg,
K _B = 1.380664 × 10 ^-23 Ws / grd is the Boltzmann constant,
T is the temperature of the cooling gas or the wall temperature in K,
X (R) is the condensation coefficient,
Δ E (R) is the activation-dependent activation energy in kcal / mol,
R is the degree of coverage,
R = 1.9872 x 10 ^-3 Kcal / mol · K is the general gas constant.

The degree of coverage can be expressed by the expression

describe.

n = n _a is the currently considered number of occupied detention sites in atoms / m².
n _m = n ₀ is the number of available adsorption sites, which does not have to agree with the number of surface atoms, in atoms / m². [5]

For the desorption rate is used after Literature [4] the term

with f ' (R = R.

∂ (R) is the desorption coefficient in atoms / m² · s.
Δ E _d (R) is the desorption in Koal / mol.

Assuming that the condensation coefficient X (R) and the activation energy Δ E _a (R) are independent of the degree of coverage R , the adsorption of a particle is in no way affected by the presence of other adsorbed particles. Under this condition, both mass flow densities or velocities for adsorption and desorption can be equated for the derivation of the Langmuirischen adsorption isotherms. In equilibrium we obtain for r _a = r _d , from the expressions (2.1.3) and (2.1.8), the equation

For b results from the transformations of the expression

With the equation (2.1.7) one obtains from the expression (2.1.9) the Langmuir adsorption isotherm, according to literature [4],

The formula for calculating the density, for the fission products on the material surface, can be calculated from the equation (2.1.9) derived. By reshaping you get first

Using the expression (2.1.7), we get

Is the occupancy of the surface by the total mass of all Looking at particles on the wall, the result is the the following equation for the wall density, according to literature [2],

N _w = K * M / (1 - M / M _s ) (2.1.12)

M _s is the total mass of particles per unit area associated with the monomolecular occupation n _m , which are present in a monolayer, in kg / m².

The vapor pressure over the material surface is determined by the equation

= K ₁ · e ^{- Δ H / RT} (2.1.13) [2]

specified.

Δ H is the heat of condensation in Koal / mol.
K ₁ is a pressure constant in N / m².

The equation (2.1.12) is used to calculate the density N _i and the mass M , the deposited particles on the surface, in the system of equations (2.1.1) and (2.1.2).

The expression (2.1.12) results from the differential equation (2.1.1)

The differential equation for calculating the mass at which Occupation of the wall surface with fission products, reads because of (2.1.12)

The coupled system of differential equations (2.1.14) and (2.1.15) can in part be simplified. In literature [7] the time dependence for N at a constant volume is neglected. For the numerical solution method the equation system then results from (2.1.14) and (2.1.15)

N ₀ is the initial density when the fission products enter the experimental setup at x = 0.

A comparison of calculated and measured results can be found in the literature [2].  

2.2 Model of Iniotakis

The model of Iniotakis is detailed in Literature [3], with all occurring individual processes in the fission product deposition, described. To later the differences between the models clearly to do is to limit oneself to the necessary equations, which also in literature [8] and [9] in the development of the Computer programs for fission product deposition have been used are. In the model of Iniotakis one also looks here, as in the Model of Neill and Kress, the condensation of fission product nuclides on the material surface and in addition one Diffusion of fission products in the wall material by a penetration process using a penetration coefficient is described. The reverse process, the process of pentration, is the Evaporation of the cleavage product nuclides from the capillary system of the wall material.

The balance equation, for the concentration of a considered Sort of fission product nuclides, taking into account the Convection, mass transfer and radioactive particle decay, is according to literature [8]

x is the location for the concentration in the direction of flow in m.
v is the mean flow velocity of the cooling gas He and of the vapor for the fission product nuclides in m / s. (∇ n is the change over time of the concentration by convection.
n (x, t) is the nuclide concentration of a variety of fission products in the cooling gas in atoms / m³.
n _w (x, t) is the cleavage product concentration for a variety of nuclides in the material wall (metal surface) in atoms / m³.
H = Sh · D _G / d _H is the mass transfer coefficient in m / s.
Sh is the Sherwood number.
D _G is the gas diffusion coefficient for the gas diffusion in the boundary layer above the material surface in m² / s.
d _H is the hydraulic diameter of the experimental setup in m.
V is the gas volume of the cooling gas per unit length in m².
O is the geometric surface of the flow channel per unit length in m.
λ is the decay constant of the nuclide in 1 / s.
λ n is the proportion of the concentration that passes through radioactive decay into other nuclides.

The following Fig. 2, from literature [8], is intended to illustrate the processes over a wall surface in fission product deposition clarify. According to literature [8], one can assume the following assumptions:

Zone I:
In the area of the cooling gas flow, the fission product concentration n (x, t) is assumed to be constant.
Zone II:
In the area of mass transfer, the concentration should change linearly.
Zone III:
In the area of the boundary layer, with the thickness of the free mean path length Λ , the concentration n _w (x, t) should be.

In equation (2.2.1), convection is the mass flow   of the cooling gas constant. Due to the cooling of the cooling gas, by delivery of Heat energy to the material wall, the density changes and the flow velocity. The mass flow for the cleavage products changes through Mass transfer and radioactive decay.

For the balance equation in zone II, in which the mass transfer takes place, one can write

With

A = α · v _⟂ · n _w (2.2.4)

and

M _O  =  ·σ ·M₀_∞      (2.2.5).

σ (x, t) is the degree of coverage.
   is the roughness factor or the ratio of actual material surface to the geometric Surface.
θ (σ) is the desorption constant dependent on the degree of coverage in 1 / s.
α is the coefficient.

is the average impact velocity of the particles on the surface in m / s.
M ₀ _∞ corresponds to n _m or n ₀, the number of atoms for a monolayer on the surface, in atoms / m².
A is the fraction of the material flow density for the adsorption in atoms / m².s.
is the fraction for the material flow density during desorption in atoms / m²s.

For the processes in activated chemical processes is for the occupancy-dependent desorption constant the Elovich relationship used. According to literature [8] is the relationship

θ (σ) = θ ₀ (T) · exp (- Q ₀ / RT) · exp ((Q ₀ - Q _s ) · s / RT) (2.2.6).

For the desorption energy Q (σ) the equation applies here

Q (σ) = Q ₀ - (Q ₀ - Q _s ) · σ (2.2.7 [8]

Q ₀ is the sorption energy in Koal / mol.
Q _s is the sublimation energy in Koal / mol.
θ _o = ν is the frequency of the lattice vibrations in 1 / s.

The following abbreviation is used in equation (2.2.4)

The temporal change of the location-dependent degree of coverage is through the balance equation

D = (λ + θ (σ)) · σ (2.2.10) [3] [8]

and

detected.

the time change caused by the adsorption is in 1 / s.
D is the temporal change of the degree of coverage by desorption in 1 / s.
β has the meaning of an adsorption coefficient here.

Compared to the penetration coefficient (1 - b) , β is the probability that the true particles on the surface do not penetrate into the interior of the material wall but stick to the surface of the material.

In order to obtain a coupled differential equation system for a numerical solution, the equation (2.2.2), using (2.2.3) and (2.2.4), is solved for n _w and one obtains first

According to literature [8], the size M ₀ has the same meaning as M ₀ _∞ according to literature [3].

M ₀ corresponds to the assignment for a monolayer in atoms / m².

The expression (2.2.12) is transformed into the equations (2.2.1) and (2.2.9), again using the formulas (2.2.10) and (2.2.11). By reshaping obtained according to literature [8] the Differential equations

The numerical solution method is given in literature [8].  

The existing penetration coefficient (1 - β) , in Equation (2.2.14), is the probability that the impinging particles will penetrate the wall material. The additional evaporation process of penetrated particles from the interior of the wall during desorption can be found in the 1st term on the right-hand side in equation (2.2.14).

In the model of Iniotakis is here after literature [10] the adsorption isotherm

used.

A is the atomic weight.

The ratio n _w / M ₀ is defined in equilibrium at constant temperature (wall temperature in K) as Adsorptionsisotherme. In a comparison of the models of Neill and Kress and Iniotakis, one considers the ratio in the evaluation of the reactor experiments Vampyr V07 and sapphire 05

as an "isotherm", it being atM _D  around the wall material diffused concentration should act. According to literature [10], this ratio depends on the measuring time, the mass transfer coefficientH, the desorption constantθ. the accommodation coefficientα, the penetration coefficient (1 -β) and the decay constantλ from. The relationship  is characteristic of every experiment and it becomes even after literature [10], as free parameter, no physical importance attached. The remarks after literature [10] will be clear later Why is there a procedure for fixing the Has developed parameters.

For the calculation of the penetration coefficient is from Iniotakis given the following equation

α is here the accommodation coefficient.

For the penetrating into the wall diffusion current I _D , the Fick's Law

used. The equation (2.2.18) is in cylindrical coordinates considered.

Φ (x, r, t) is the amount of fission products which diffuses into the wall material at the point (x, r) at time t , in atoms / m³.
D is the diffusion constant for the wall material in m² / s.
I _E is the mass flow density for the incident particles on the material wall in atoms / m².s.

The formula for I is _E

M is the particle mass.
T is the gas temperature in K.

For the radius r in (2.2.18), the range R _i < r < R _a R _i and R _a are the inner and outer radii.

In equation (2.2.17), I _R is the fraction for the mass flow density when the particles are reflected on the material wall, in atoms / m² · s.

The accommodation coefficient α is given by the expression

described.

The diffusion current into the interior of the wall can be calculated from equation (2.2.18) using equations (2.2.17), (2.2.19), and (2.2.20). The penetration coefficient (1- β) is estimated from the measured value M (O, t) , the amount adsorptively bound at the position x = 0.

According to Iniotakis, the pentration coefficient should be based on  Enthalpy effects in first approximation as temperature dependent to be viewed as.

According to the explanations in literature [10], the model of Iniotakis for the case (1 -β) = the model of Neill and Be equivalent to kress. A comparison of calculated results with the Measurement results of the reactor experiments Vampyr V07 and Sapphire P05 can be found in the literature [10]. One can only assume that the calculated data with help of the free parameter  in equation (2.2.16) and μ in (2.2.15) were adapted to the measured values, until at the bills a match, by trying out the different "isotherms" in Fig. 6 in literature [10], has reached.  

2.3 The experiment VAMPYR II / V01

To determine the fission product deposits, for the materials Inconel 617 and Incoloy 800H, the reactor experiment VAMPYR II / V01 carried out at the AVR in Jülich was used. The experimental set-up used was a rod made of Inconel 617 and a test tube made of Incoloy 800H, both of which were assembled into an annular gap arrangement. This annular gap arrangement was flowed through at a test duration of about 60 days with the diverted primary gas flow of the HTR. In this case, the incoming cooling gas was heated to an inlet temperature of about 810 ° C when entering the experimental setup, so that one reaches approximately the conditions in which the metallic reactor components are exposed to the cooling gas flow in the primary circuit. At the end of the experiment, the activities of the fission product deposits for each nuclide were measured for each segment of the array using γ spectroscopy. From the measurements of the individual segments, a profile of the activity distribution for each individual nuclide can be plotted as a function of the length x and of the temperature profile of the cooling gas in the experimental setup. The measurement results can be found in Section 7 and taken from the literature [25].

For the later invoices are the following technical information and physical data of importance:

Inner diameter of the test tube made of Incoloy 800H | = 0.014 m, Outer diameter of the test rod made of Inconel 617 = 0.009 m, free flow area = 9.032 x 10 ^-5 m², Length of the test tube = 3.62 m Length of the test bar = 3.59 m, hydraulic diameter d _H = d _pipe - d _bar = 0.005 m, Gas pressure in the primary circuit = 10.8 bar, Mass flow of helium = 2.446 x 10 ^-3 kg / s, Inlet temperature of the cooling gas = 810 ° C, Outlet temperature of the cooling gas ≈ 418 ° C.

From the measurements arise at closer inspection for the activity distributions four course options, the for the activities on the material surfaces in question come.

The curves are shown in the following Fig. 3. For the behavior of the activities, the curves in Fig. 3 should be briefly explained.

Curve 1 :
The activity of a nuclide, on the surface in the experimental setup, decreases with decreasing gas temperatures.
Curve 2 : The activity of a nuclide increases with decreasing gas temperatures.
Curve 3 :
As the cooling gas temperatures decrease, the activity increases, as in curve 2 , but the slope of the increase is different.
Curve 4 :
The activity on the material surfaces increases up to a maximum and falls off again as the gas temperatures decrease.

Curve 4 has been observed only for the nuclide Ag-110 m, and the remaining traces are found in both the metallic and non-metallic fission products.

2.4 Evaluations of the activity distributions from the Reactor experiments with the models used so far

The only way to study the physical processes has so far available, is the Recalculation of the measured activity profiles with the help of a the models used.

In the recalculations, the parameters and their associated Constants adapted as long as you can at your discretion decides if the calculated profiles of the profiles work well with match the measured activity profiles. In this method, there is the difficulty that the selected Parameters when using different computer programs, with different solution methods, different values and thus a comparison of the data sets Alone complicated for a single reactor experiment becomes.

When using the SPATRA program the KFA / ISF turned out that the application of the model of Iniotakis itself only calculate the measured activity curves of Ag-110 m to let.

A table summarizing the results you get taken from the literature [11], can be found in the Section 7.

Despite predetermined data sets, such as penetration coefficient and desorption energy, was found in the other nuclides, z. Curve 4 according to Fig. 3 is always shown for Cs-137 . For Cs-137 the result for Inconel 617 is Curve 1 and for Incoloy 800H for Curve 2 . In order to use the model in Section 2.2 to recalculate the measured curves of Cs-137 with the SPATRA program, the penetration coefficient (1 - β) in equation (2.2.14) was changed to higher values and the desorption energy.

When increasing the penetration coefficient is achieved at an increase in nuclide concentration in the Interior of the wall material and a simultaneous decrease the nuclide concentration for the fission products on the Material surface.  

An increase in the desorption energy shifts the position of the maximum in the curve 4 , in Fig. 3, to the left to higher temperatures, so that one can approximate the curve 4 to the curve 1 .

When the desorption energy is lowered, the maximum migrates to the right to smaller temperatures and one approaches with the curve 4 of curve 2 . In this type of evaluation of the reactor experiments, one may expect a higher desorption energy in the course of curve 1 , at Cs-137, but in the course of curve 2 , it may be in the reduction of the desorption to contradictions with the actual values Laboratory tests come.

Due to the length of the experimental setup in Fig. 3, it can not be decided at reduced desorption energy whether the course of the calculated activities at lower temperatures is justified for curve 2 . The program SPATRA of the KFA / ISF was only applicable to the nuclide Ag-110 m, because for the curve 4 , in Fig. 3, an additional desorption is recognizable. For the other nuclides, there was no clear physical explanation for using these evaluation methods. The limited application of the SPATRA program may be due to the equation (2.2.12) used in the construction of the coupled system of differential equations (2.2.13) and (2.2.14).

One is always on the model of Iniotakis of the assumption assumed that all nuclides desorption in the Spaltproduktablagerung must be present.

Experts have already speculated that the Spaltproduktnuklide be bound to dust particles and thereby a different adsorption and desorption behavior show on the metal surfaces.

One can counter this assumption, if the Concentration of the dust particles is small and the nuclides in the gas phase as an ideal gas, then you can no interactions with the dust particles in sufficient Measurements take place and not even when the nuclides on the material surface are bonded to the dust particles.  

The measurements of Ag-110 m show that the possible sites of adhesion are not covered in sufficient numbers with dust particles, especially Ag-110 m compared to the other nuclides a very has small partial pressure and thus in the gas state a has lower particle number.

If the dust as a possible cause of another behavior the activity distribution is considered, then it should be at the nuclide Ag-110 m also give no detectable desorption, the measurements do not confirm this hypothesis. On another possible cause of the behavior of the Activity distributions are discussed in sections 3 and 7.

Another evaluation of the VAMPYR II / V01 experiment is to be found in the literature [12]. The calculations here with the evaluation program used RADAX were performed in [12], unlike the Program SPATRA at Ag-110 m no indication of any Recognize desorption in the calculated curves. For the calculations for Cs-137, for Inconel 617 and Incoloy 800H, the differences between measured and calculated activity almost an order of magnitude, especially the Slopes of gradients for Inconel 617 and Incoloy 800H, with records of HRB and KFA, opposite to the measured values. In addition, the calculated curves were determined from Temperature ranges extrapolated. If, according to the statements in literature [10], it is said that the agreement between theory and experiment is very good, then you can not understand the results in literature [12] if in the evaluation in [12] the data sets of the KFA have been used.

At first there are only two explanations, once you can in the program RADAX another solution method than in the Program SPATRA have been used or the records from KFA could be wrong.

The images and results from [12] are in the Data in section 7, comparing the different results, with included. In a valuation of the calculated results in literature [12] is itself acknowledged that for temperatures above  600 ° C the penetration coefficient for Cs-137 uncertainties having.

According to HRB evaluations, the uncertainties around the Factor 2 to 4 up and down by a factor of 10 be reduced.

At temperatures below 600 ° C, large uncertainties in the temperature dependence of the desorption constants occur for Cs-137. For the nuclide Ag-110 m, a new definition of the deposition parameters is required in literature [12]. The desorption energy should be changed from 30 to 67 Kcal / mole and the penetration coefficient (1 - β) should be changed from 2 · 10 ^-4 to 2 · 10 ^-6 .

From the results in the literature [11] and [12] becomes clear that you are using the model of Iniotakis not only the difficulties in setting the parameters, but also in the physical interpretation of the Has results.

2.5 Criticism of the state of play in determining the Cleavage product deposits

The criticism, which hereby from the explanations in the Section 2.4 gives less focus on the models, but on the verifiability of the model in section 2.2. From the previous publications of the data for VAMPYR II / V01 experiment are just the surface activities known.

From these measurements can be in the reactor experiments no Difference in the concentrations of fission products the material surface and the concentration inside the Wall material are determined.

As long as no depth profile measurements, z. For example, by removing the upper layers of material to measure the activity in each layer, the use of the model in Section 2.2 will always cause difficulties. In the model of Iniotakis one needs to determine the penetration coefficient (1- β) the parameter α, the concentration n _w , which is estimated by M (O, t) for the determination of I _E from the measured values at x = 0, the material current density I _D , which in Equation (2.2.18) depends on a temperature-dependent material diffusion constant. Considering the fact that the material diffusion constant also depends on the degree of corrosion of the material, then arise in the definition of the data sets in the individual investigations of the physical and chemical processes involved significant problems.

Explanations of diffusion processes of fission products and Particles in oxide layers can be found in the literature [3]. If at the carried out reactor experiment already before Beginning of the measurements the corrosion degrees of the used Materials have not been studied, then the model from Neill and Kress much easier on a review of measurement and calculation results. The parameter derived from the measured activities by the Agreement of the calculated activity profiles determined can be, is the constantK in equation (2.1.12). The parameters for the roughness of the surfaces and the Number of detention places is needed on all models. In the model of Neill and Kress it is the sizeM _s  and in the model of Iniotakis it is the parameters  andM₀_∞, These sizes are going through for different materials Measuring methods of electron and ion emissions Material surfaces determined with adsorbed particles are occupied. [13] [14]

The determination of the surface roughness takes place after BET method. [15]

These experiments are only used as laboratory experiments with a Variety of particles carried out in the gaseous state, the then adsorbed by the surface of the material. Reactor experiments have about 260 different nuclides present in the cooling gas stream, resulting in the adsorption possibly influence each other. Use of the determined parameters from laboratory experiments can be used in the evaluation and investigation of fission product deposits questionable in reactor experiments.  

3. Modified model for the description of fission product deposition on metal surfaces

Before this model is developed from the two models in section 2.1 and 2.2, it is first necessary to find an equation for the material flow density of the adsorbed particles from the evaluations of the reactor experiment, which is true for all possible Spaltproduktnuklide. From Fig. 3 it is clear that only the curve 4 is due to the net rate of the two mass flow densities of adsorption and desorption.

The curve 1 behaves like the course of the concentration in the "model of the black wall", where all particles are adsorbed and no desorption is present. The possible course types in Fig. 3 indicate that the interactions of the fission product nuclides with the material surface are the cause. This interaction must be taken into account in the adhesion coefficient a and s in equation (2.1.4) in section 2.2 so that the curves 1 to 3 in FIG. 3 and the course 4 with additional desorption can be obtained.

3.1 Equation for the mass flow density during the adsorption process the cleavage product deposits

In the evaluation of the activity distributions, it is assumed that curves 1 to 3 , in FIG. 3, are determined only by adsorption processes and that the desorption processes can be greatly reduced or no longer occur. [11]

To determine a term for the adhesion coefficient, may it can be assumed that the Hertz-Knudsen formula in [15]

is applicable.

If ≈ 0 then the desorption should be negligible and you get the equation (2.1.3)

is the vapor pressure associated with T in N / m².
J is the number of atoms that emanate from the surface per unit of time in a time t , in atoms / m² · s.
J ↓ = r _a is the mass flow density for the adsorption of the particles in atoms / m² · s.
Z is the number of impacts according to equation (2.1.5) in atoms / m² · s.
T is the cooling gas temperature in K, which corresponds to the surface temperature of the pipe inner wall. [16]
p _d is the partial pressure of nuclides in the cooling gas, which can be calculated from the gas equation for ideal gases, in N / m².
can be determined from the expression (2.1.13) if additional desorption occurs.
α = s is the adhesion coefficient to be investigated.

In order to obtain an expression for α or s from equation (3.1.2), one needs the partial pressure of the nuclides, which can be estimated from the measured hot gas activities for the individual fission products in the high-temperature reactor. To calculate the partial pressures from the hot gas activities, one can use the following formula

p _d = 1.98661 · 10 ^-22 · p _He · x _A · τ _1/2 p _d in N / m² (3.1.3) [11]

use.

x _A is the measured hot gas activity in 10 ^-12 Ci / m³.
p _He is the operating pressure of the cooling gas He, in the HTR, in the unit bar.
τ _1/2 is the half-life of the nuclide in s.

Table 5.1 in [11] shows in Table 1 the specific Hot gas activities at a cooling gas temperature of 850 ° C for the estimation of the partial pressures with the help of Eq. (3.1.3) used because these values at an inlet temperature of 810 ° C, according to Section 2.3, the experimental conditions best correspond.

The actual hot gas, when performing of the VAMPYR II / V01 experiment, were in the Publications of the results are not indicated, because the required source strengths, to calculate the initial concentration for the models, from the measured Filter activities were determined.  

Table 1

Specific hot gas activities for the nuclides at a cooling gas temperature of 850 ° C at AVR Jülich

For the estimation of the partial pressures from Table 1, with Equation (3.1.5), the following values result in Table 2:

Used partial pressures in the evaluation of the VAMPYR II / V01 experiment nuclide Partial pressure p _d in N / m² Cs-137 | 2,713 · 10 ^- ⁸ Cs-134 6,987 · 10 ^- ¹⁰ Eu-154 3.033 · 10 ^- ¹⁰ Eu-155 2,466 · 10 ^- ¹⁰ Co-60 8,341 · 10 ^- ¹⁰ Zn-65 2,705 · 10 ^- ¹⁰ Fe-59 7.613 × 10 ^- ¹¹ Cr-51 3,904 · 10 ^- ⁹ Sc-46 2.418 × 10 ^- ¹⁰ Ag-110 m 6.890 × 10 ^- ¹¹

With the help of the partial pressures one is capable of the shock number from the equation (2.1.5).  

The activation energy Δ E _a required in equations (2.1.3), (3.1.2), and (2.1.4) is determined from an e ^x -fit by taking equation (3.1.2) as an equation the form

considered.

The equation (3.1.4) is based on the assumption that the decrease of the activity distribution to lower temperatures is determined by the Boltzmann factor. The time derivative in (3.1.4) is replaced by the estimate d n _a / d t ≈ n _a / t _B and n _a can be derived from the radioactive decay law

determine.

According to Section 2.3, the measurement time of the experiment t _{B is} 60 days or 5.184 · 10⁶ s. If one reduces the equation (3.1.4) to a linear regression line, one obtains the equation

ln (n _a / t _B ) = ln a ₀ + bx (3.1.6).

Here, b = -Δ E _a / R and x = 1 / T. R is again the general gas constant in Kcal / K · mol. Even if the exact value of the time derivative d n _a / d t is not available, the slope in (3.1.6), which is determined by the Boltzmann factor, should not change significantly, but the change takes place in the pre-factor a ₀.

Applying the equation (3.1.6) to all activity curves for the nuclides one after the other, the determination of the constant b is only physically meaningful if b is 0.

If the condition b is 0, then the Boltzmann factor is 1, because it must not be greater than 1.

If, in the application of equation (3.1.6), the Boltzmann factor is 1, the following results are obtained for the course of curve 1 , in FIG. 3, which are listed in table 3.

Table 3

Determined activation energies and pre-factors for the activity distributions according to the curve 1 in Fig. 3

The negative sign in Tab. 3, in the activation energy states only that the energy is released during the adsorption of the particles. The sign for the activation energy has nothing to do with the sign of the constant b , because the sign for b itself must result from the equation (3.1.6), considering n _a / t _B as a function of the temperature of the pipe inner wall.

In the further evaluation, there are the following possibilities:

• 1) one determines the adhesion coefficient α from the equation
• 2) Taking into account the activation energy, the equation results

Between α ₁ and the temperature T , the following relationship with the degree of coverage σ can be produced

The exponent m and the constant C can be calculated from the reduced regression line of equation (3.1.9). The equation for the equalizer line has the shape

log₁₀ (α ₁ / σ) = log₁₀ C + m · log₁₀ (T / T ₀) (3.1.10).

T ₀ = 273.15 K is an arbitrarily used reference temperature so that the adhesion coefficient has no physical unit.

For the calculation of the degree of coverage σ = n _a / n ₀, the values n ₀ for the individual nuclides, the possible sites for a monolayer in atoms / m², had to be calculated from the ratio

n ₀ = A ₀ / r _A ² · π

be estimated.

A ₀ is the area of 1 m² and r _A is the atomic radius of the nuclide.

The values for the atomic radii, which were determined using the ordinal numbers from the literature [17], are compiled with the calculated values for n ₀ in Table 4.

Table 4

Estimated number of atoms for a monolayer as a function of atomic number

For the constant C and the exponent m , the following results are obtained from the regression line (3.1.10), in Table 5:

Table 5

Data for the calculation of the adhesion coefficient for the equation (3.1.12)

From the equation (3.1.9), one obtains with the expressions (3.1.7) and (3.1.8) for the coefficient of adhesion following equations:

• 1) with equation (3.1.7) results
• 2) with equation (3.1.8) is obtained

Equation (3.1.11), in [11], is a special case of Equation (3.1.12) and also shows physical properties related to equilibrium statistics. Looking from the literature [6] the material flow density for the particles, which impinge on the wall surface in a solid solid angle 4 π , then one first obtains the equation

With

from the equation for ideal gases

Taking into account a sticking coefficient, you get the material flow density for the adsorbed particles in the shape

If n ~ n _a or n ~ σ = n _a / n ₀, then the temperature dependence is due

in equation (3.1.11) of the Dependence in  =v _⟂ ago. From equation (3.1.14) one then obtains forα the expression

The equation (3.1.15) then corresponds to the equation of temperature (3.1.7) in the investigation of the temperature dependence. From the two equations (3.1.7) and (3.1.15), a ~ is when there are no additional temperature dependencies due to the influence of the solid-state lattice and the adsorbed particles on the material surface. The equation (3.1.8) goes into the expression (3.1.7) or (3.1.15) if Δ E _a = 0. The same dependencies mentioned here also appear in equation (2.2.4).

The equation for the material flow density in the adsorption process has the form because of the expression (3.1.12) and (3.1.2)

The constant C corresponds to the condensation coefficient (R) in equation (2.1.4). The function f (R) = (1 - R) = 1 - σ) is in equation (2.1.6) f (R) = f (σ) = 1, if the degree of coverage σ «1.

Since one knows the equation for the mass flow density in (3.1.16), one must show that the relation (3.1.16) is also applicable to the curves 2 and 3 , in Fig. 3, so that the Boltzmann factor exp (- Δ e _a / RT) is 1, as required by the equilibrium statistics. The equation (3.1.10) and (3.1.6) satisfy this condition for the Boltzmann factor only for the curve 1 and is therefore unsuitable for the courses 2 and 3 . Although one can not yet consider the mass transfer in these evaluation methods, one nevertheless already obtains values for the Boltzmann factors 1 if one extends an equation of the form (3.1.10) or (3.1.6) to several variables.

With the help of equation (3.1.16) one can use the following linear Equalization line, with several variables after the Method of least squares, by

receive.

Here, the number of bumps in the left term in (3.1.17), so that the straight-line equation only Has 3 variables. One can see the equation (3.1.17) in the form

U = a ₀ + a ₁ X + a ₂ Y + a ₃ Z

or after [18] in the form

X ₁ = a ₀ + A ₁ X ₂ + a ₂ X ₃ + a ₃ X ₄ (3.1.18)

Write if you use the following abbreviations:

U = X 1 = ln ((d n _a / d t) * t) *),
X = X ₂ = ln (T / T ₀),
Y = X ₃ = ln σ = ln (n _a / n ₀),
Z = X ₄ = 1 / T.

With the constants a ₀, a ₁, a ₂ and A ₃, the values for the constants C, m and the activation energy Δ E _a are obtained from the fit. The constants correspond to (3.1.17) the following expressions:

a ₁ = m , a ₂ = 1, a ₃ = - Δ E _a / R.

For C and Δ E _a , the upper abbreviations result

and

Δ E _a = a ₃ · R

The Boltzmann factor is again 1 when the constant from the fit a ₃ is 0.

The following table should, according to the designations in the straight line (3.1.18), the names of the Correlation coefficients and the relationships of the data reproduce that one for the results in the tables Tables 7 and 8 are needed.

Table 6

Name of the correlation coefficients and the relationships between the variables in Eq. (3.1.17) and (3.1.18)

The multiple correlation coefficient R _1.234 = R _U.XYZ is obtained from the relationship

R _1.234 = (1 - (1 - R ¹²₁₂) · (1 - R ² _13.2 ) · (1 - R ² _14.23 )) ^1/2 [18]

calculated.

The formulas for the calculation of the partial correlation coefficients can also be found in the literature [18].

The results for the determined constants and Correlation coefficients from the equation (3.1.17) or (3.1.18), for the individual nuclides, are for the VAMPYR II / V01 experiment in Tables 7 and 8 compiled.  

Table 7

Values for C, m and Δ E _a from the regression line (3.1.17) for all measured nuclides in the VAMPYR II / V01 experiment.

Table 8

Correlation coefficients for the equation (3.1.17) in the evaluation of the profiles of all measured nuclides in the VAMPYR II experiment

From the values of m can be seen that a majority of these exponents are at 0.5, when the activation energy Δ E _a are very small. The adhesion coefficient depends on the temperature behavior only on the impact velocity of the particles. For larger activation energies, the values for m are less than 0.5. These deviations could be interpreted so that the heat of the fluid or the cooling gas is released to the material wall and thus the lattice vibrations are affected. In addition, with sufficient activation energies, the released heat can also be absorbed by the solid-state lattice. Also, the accumulating particles can cause a change in the specific heat and thermal conductivity by changing the lattice vibrations on the material surface. Although one can not find out the details of the possible causes with the results in Tables 7 and 8, it turns out that the partial correlation coefficients R ₁₂ = R _UX , the relationship between the material flow density and the factor (T / T ₀ ) ^m , for amounts above 0.5 indicate a possible influence of temperature in equation (3.1.16). This conclusion follows from the condition that the Boltzmann factor must be 1. The correlation coefficient R _14.23 = R _UZ.XY shows the relationship between the material flow _density and the Boltzmann factor.

With the equation (3.1.16) it is possible to describe all types of events in Fig. 3, without having to consider all the individual material properties and interactions of the different nuclides in equation (3.1.16). It has the advantage that the equation (3.1.16) is applicable to all nuclides, as far as can be seen from the VAMPYR II / V01 experiment, and no special equations for the material flow density in the individual fission products are needed.

With the method in Section 5, the constants C, m and Δ E _{a are determined} from the measurements of the activity distributions, taking into account the mass transfer, so that the type of course in Fig. 3 is determined by the values of the three constants.

The correlation coefficient R _13.2 = R _UY.X = 1 states that the mass flow density is proportional to the number of adsorbed particles according to Equation (3.1.16) or (3.1.11) and (3.1.12).

Between the exponent m and the activation energy Δ E _a , for Inconel 617 and Incoloy 800 H, connections can be established, which can be deduced from the fits in the equations (3.1.6) and (3.1.10) as well as from the equation (3.1.17) result. If one plots the values for m as a function of the activation energy, then, according to FIG. 4, two straight lines appear which are characteristic of the two materials. The straight-line equation obtained from the values m and Δ E _a are for both materials:

for Inconel 617
m = .4764 to .4869 · Δ Δ E _a E _a in Kcal / mol (3.1.19)

for Inconel 617
m = 0.4999 to 0.5838 · Δ Δ E _a E _a in Kcal / mol (3.1.20).

From Fig. 4 it is seen that the intersection of the two degrees at m = 0,5 must lie when Δ E _a = 0. At the intersection in Fig. 4, the conditions for the temperature dependence of the adhesion coefficient α in equation (3.1.11) are present. By means of the equations (3.1.19) and 3.1.20) one can reduce the equation (3.1.16) to the two parameters C and Δ E _a . This simplification could be advantageous if the constants C and Δ E _a , as described in Section 2.1, as coverage-dependent parameters.

If it increases with occupancy of the material surface with fission product nuclides to saturation effects Adsorption process should come, so can for the Stoffstromdichte tentatively the approach

for f (σ) = f (R), the equation (2.1.6) can be used.

In the evaluations of the VAMPYR II / V01 experiment, due to the small degrees of coverage, no hint of incoming saturation effects could be found in (3.1.21). Therefore, in the further considerations, we will consider equation (3.1.16) instead of equation (3.1.21). The expression for f (R) may depend on the properties of the solid-state lattice and must then be considered in (3.1.21) if the adsorption process becomes dependent on it.

3.2 equations for the modified model

In order to calculate the concentration of fission products in the cooling gas flow, one can assume the equations (2.1.1) and (2.2.1) in the approach to the balance equation. Instead of the isotherm, to describe the interaction of fission products on the material surface with the fission product nuclides in the gas phase of the cooling gas flow, which is required to calculate the wall concentration in the occupation of the wall surface with fission products, the equation (3.1.16) is used in connection with Equation (2.1.8). If we consider in the first term on the right side in equation (2.2.1) the quantity Hn _w , then it is the mass flow density that corresponds to the material flow density for the net rate from the two mass flow densities for adsorption and desorption. The isotherm can then be through the equation

Hn _w = A - D (3.2.1)

be replaced. If, taking into account the particle decay, equation (3.1.16) and equation (2.1.8) are used, the two equations for the material flow densities A and D are obtained

With

δ = n · n _d (3.2.4).

n _a is again the number of atoms that occupy the adhesive sites on the wall surface at a given time, in atoms / m².
n _d is the number of atoms that can leave the sites at a given time by desorption, in atoms / m².
ν is the oscillation frequency of the solid lattice perpendicular to the wall surface in 1 / s.
δ is again the desorption coefficient in atoms / m² · s.

If Hn _{w is taken to} be the net rate of mass flow densities of adsorption and desorption then the partial differential equation follows for the balance equation of the fission product concentration

O / V is again the ratio of cooling gas volume per unit length to the geometric surface of the flow channel per unit length.

In the equation (3.2.1) or (3.2.5), depend on the nature of the interaction of the Schaltproduktnuklide on the surface of the nuclides in the gas phase only of the constant or of the parameters C, m, Δ E _a, and Δ E _d from, which can also be dependent on the degree of coverage with increasing occupancy.

As already indicated in section 2.5, the Roughness of the surface by laboratory tests with the help of BET method [15] determined.

Under existing conditions in reactor experiments, the surface roughness could also change with increasing occupancy of the surface with different fission products. Another difficulty arises when, after completion of a reactor experiment, the surface roughness of radioactive material surfaces should be examined. It is advantageous to be able to circumvent the use of a roughness factor for these reasons alone, because the laboratory tests in the BET method can contain uncertainties in the measurement results. It is possible to find another parameter that can be determined from the activity distributions by appropriate choice and indicates the fraction of mass flow density that represents the excess concentration over the boundary layer. If the roughness of the surface is large, then it is to be expected that more particles will be able to find their sites on the material wall during mass transfer, and thus fewer nuclides will leave the flow channel once the desorption is considered separately from the process. The proportion of the material flow density, which consists of the concentration excess over the concentration in the boundary layer, can be expressed by a multiple of the net rate (AD) . Thus, equation (2.2.2) takes the form

- H (n - n _w ) = η · (D - A) (3.2.6)

or

- H (n - n _w ) = η · (A - D) .

The negative sign means that mass transfer occurs from the higher concentration areas to the lower concentration areas. If η = 0 then H · n = Hn _w and the particles would all accumulate on the material surface as in the "black wall model", if only the adsorption process is considered. The case η = 0 would only be present if the material surface were able to take up the net rate of particles from the gas phase, if the concentration n is correspondingly low.

As with the model in section 2.2, the degree of coverage of the Adsorption and desorption be dependent. According to the remarks in literature [19], here too, by equation (3.2.1), for the calculation of the Degree of coverage the net rate of adsorption and Considered desorption current density. The equation (2.2.9) is then given by the equation (3.2.1) the figure

n ₀ is again the possible number of detention sites for a monolayer in atoms / m².

In order to obtain a system of differential equations, the equation (3.2.6) is solved for the mass flow density A and it follows

Here one has for the equation (3.2.8) in the relationship (3.2.6) uses the expression (3.2.1).

Figure 5 illustrates the operations in a volume element considered in this model.

Substituting equation (3.2.8) into expression (3.2.7), then results for the degree of coverage

For this modified model you get the following Differential equations

For the material flow densities A and D , other terms may be used. The terms used herein are useful in calculating the activity distributions in the VAMPYR II / V01 experiment.

Considering the equation (3.2.1), C ~ H can be. To study the temperature behavior (T / T ₀) ^m in equations (3.1.16) and (3.2.12), the following dependencies can be considered for C :

C =C₀ · C₀ in s / m (3.2.13),

C =C₀ ·  ·H C₀ in s² / m² (3.2.14),

C = C ₀ (3.2.15).

The temperature behavior for the material flow density of the Adsorption process in (3.1.16) and (3.2.12) is for the Absorbency of fission products is crucial, whether in larger or smaller temperatures on the material surface more or less cleavage product concentrations are to be expected.

In this model only, as with the model of Neill and Kress, the fission product deposits on the material surfaces considered and no additional diffusion processes through Penetration, as in section 2.2. The diffusion processes into the wall interior of the material can be considered independently by the law of diffusion, if the surface concentration after this and the Model calculated in section 2.1.  

4. Solutions from the modified model

To be able to develop the procedure in Section 5, it is it is necessary to have an analytical solution available which allows, under certain conditions, the necessary parameters in equation (3.2.12). This solution must also be understandable, so that one of Computer programs is independent.

4.1 Solution of the equation system assuming a decreased desorption

An analytic solution of the equation system, in Section 3.2, is only possible if there are no saturation effects during the adsorption of the particles. In order to solve the system of equations (3.2.10) and (3.2.11) separately, one has to separate this system from the variables n and σ . Because of (3.1.14), one can undertake the following experiment and use the approach

Thus, a proportionality between the wall concentration n _w and the concentration of n for the nuclides in the cooling gas flow is present, it is necessary to insert a proportionality factor F (C m, Δ E _a, t) = F, which contains the parameters for the properties of the material, so that the interaction of the Spaltproduktnuklide with the material surface is taken into account in place of an isotherm. With the proportionality factor F , the equation (4.1.1) first results in connection with the expression (3.2.1)

or

With reduced desorption, it follows from (4.1.2) the equation

If we replace the term (AD) with (4.1.2) in the equation (3.2.10) or (4.1.3) with a reduced desorption, we obtain a partial differential equation for the concentration n , which is independent of σ becomes. Using, for the flow in the axial direction of x , the expression

then the differential equation (3.2.10) takes the form

The partial derivative ∂ v / ∂ x can later be replaced by a function if the gradient for the flow velocity can be calculated by the cooling gas density and from the temperature profile of the cooling gas temperature. In equation (4.1.4) only the partial derivatives of n are considered. For a better overview, it is useful to use the abbreviation

to use.

For a flow in a pipe, O / V = 4 / _dH . The partial differential equation (4.1.4) simplifies too

With the solution approach

n (x, t) = X (x) * T (t) (4.1.7)

obtained with the Bernoulli separation method  from equation (4.1.8) during forming

 ·X = -K ·T ·X -v · ·T      (4.1.8)

or

 is the derivative of time, of time proportionT (t), for the concentrationn,
is the derivative afterx, the location shareX (x), for the concentrationn,

The two sides in equation (4.1.8) represent two differential equations if they are equated to a constant χ . From (4.1.8) one obtains

The two differential equations for the position and time component of the concentration n are given in (4.1.9)

 +χ ·T = 0 (4.1.10)

and

Because the mass flow , as in section 2.2, as constant is considered, one canχ =λ put. This assumption can be illustrated, if in this case the Mass transfer is not considered and after the Continuity equation of the incoming mass flow equal to the outgoing mass flow in the flow channel is. But because the particle rate can still decrease, so will the time proportion of the concentrationn only by the radioactive decay. Thus, the constantχ determinable by the decay constant.

The solutions from (4.1.10) and (4.1.11) are then

T = T ₀ · e ^{- χ · t} (4.1.12)

and

The solution (4.1.7), together with the equations (4.1.12) and (4.1.13), gives the solution for the concentration n of the nuclides, for the equation (3.2.10) or (4.1.4), in shape

N ₀ is the initial concentration at a given inlet temperature T _e at x = 0, from the time t = 0, in atoms / m².

The value for N ₀ is calculated from equation (3.1.3) from the measured hot gas activities from the equation for ideal gases.

If the partial pressure is determined from (3.1.3), then for N ₀

N ₀ = p _d / K _B T (4.1.15).

For T then the value for the inlet temperature T _{e of} the cooling gas is to be used.

N ₀ can be regarded as source strength here.

The solution for the degree of coverage is obtained from the equation (3.2.11), if for n the expression (4.1.14) is used.

Then one integrates the equation (3.2.11) with the help of the Substitution process only over time and receives for the degree of coverage the solution

For the time t = 0, the condition σ = 0 results, as it must be in numerical solution methods. In the solutions (4.1.14) and (4.1.16) it can be seen that the cleavage product deposition is determined not only by the mass transfer coefficient H and by the parameters C, m, Δ E _a , Δ E _d , but also by the flow rate of the cooling gas, in which the gap products are carried.

The proportionality factor F , which in the two solutions (4.1.14) and (4.1.16) is still undetermined by the abbreviation K in the expression (4.1.5), can be expressed by the expressions for n and σ from the equation (4.1.2 ) or (4.1.3).

The equation (4.1.3) is solved for F and for n the solution (4.1.14) is used and F is given the expression

The initial concentration N ₀ can also be obtained from the degree of coverage of the solution (4.1.16). The conversion from (4.1.16) to N ₀ delivers

By inserting (4.1.18) into (4.1.17), we get F

For d n _a / d t , we use the expression (3.2.2) in equation (4.1.19) and take into account the relation σ = n _a / n ₀. From (4.1.19), then, with (3.2.2) assuming a reduced desorption, F is followed by the expression

By the equation (4.1.20) the system of equations (3.2.10) and (3.2.11) solvable.

In an analytical solution, the factor F must be independent of the degree of coverage.

For C , one can also use equations (3.2.13) to (3.2.15) in F.

For the time t = 0, F = 0 and this means that a smaller amount of material is transferred to the material wall.

For a reduced desorption, no further parameters occur in F here.

4.2 Solution of the equation system assuming additional desorption

The calculation is as in section 4.1. In additional desorption, the equation (4.1.2) is used in determining F and the expression for d n _a / d t in equation (4.1.19) is substituted by the equation of n d _n / d t.

If equation (3.2.12) is used in (4.1.19), then taking into account σ = n _a / n _o for F yields the equation

With additional desorption, the parameters δ = n _d · ν , n _d / n _a appear in F.

The parameter n _d can be replaced by the parameter n _a . If, according to literature [15], the probability w for leaving the surface of a particle after 1 second

is, then n _d = n _a , if one calculates the number of desorbed particles per second, because the product of the particle number n _a with the probability w must give the number of desorbed particles per unit time.

The parameter n _d only shows that for n _d ≈ 0 the model can be converted into an adsorption model for the curves 1 to 3, according to Fig. 3, if one has no exact clues for the desorption energies due to the sublimation energies.

When n _d = 0, it must be remembered that with a reduced desorption a change in the parameters C, m, Δ E _a and η is to be expected.

For large decay constants λ = χ , numerical difficulties can arise in the e-function e ^{χ t} at long times t . A range of application for the solution can be estimated from the e-function.

Considering in (4.2.1) n _d = n _a , the expression for F simplifies

The parameter δ then depends on n _a and is (4.2.2) δ = n _a · ν .

By means of expression (4.2.2), it is possible to look at the calculations for Ag-110 m show that it according to the statements in Section 2.4, in the case of changes in literature [12], is possible is that the desorption energy near the Sublimation energy of Ag-110 m at 67 Kcal / mol must be.  

4.3 Numerical solution for the system of equations

In the event that the constants C, m , Δ E _a and Δ E _d are dependent on the degree of coverage, if the occupancy of the surface increases and possibly also saturation effects could occur in the adsorption process, the equation system (3.2.10) and (3.2. 11) are solved numerically. Since the equation (4.1.14) is a solution of the equation (3.2.10) or also (4.1.4), one can attempt to transform the partial differential equation (4.1.4) into a differential equation that depends on the concentration n and is dependent on the time t in the calculation process.

The partial derivative ∂ n / ∂ x in (4.1.4) is replaced by the expression by differentiating the solution (4.1.14) into x .

For the partial derivative of the concentration n we obtain (4.1.14)

It is assumed that the gradient of the flow velocity ∂ v / ∂ x , in a linear temperature profile in a pipe, no longer depends on the location x and thus remains approximately constant due to the density change. An estimate for the gradient of the flow velocity can be found in section 6.

In other applications, where ∂ v / ∂ x is not constant, the exponent in (4.3.1) should be derived again after x according to the chain rule.

With the equations (4.1.6) and (4.3.1) one obtains for the Equation (4.1.4)

The degree of coverage is given in (4.3.2) in the abbreviation K in the expression for F.

If, according to equation (3.1.21), an additional saturation effect would have to be taken into account, then the expression for F can be derived from the equation (4.1.19). If the parameters in F can also depend on the degree of coverage, then (3.1.21) results from (4.1.19)

For f ( σ ) = f ( R ) the equation (2.1.6) can be used. If the adsorption is dependent on the structures of the solid lattice, then other expressions can be used for f ( σ ) .

When using a calculation method, equation (4.3.2) specifies the location x and the temperature T , which are present in a segment of the flow channel.

The time t and the degree of coverage σ are then, in the calculation of the concentration n, the variables which are needed in the approximation method for solving the differential equation.

The degree of coverage then becomes again from the equation (3.2.11) calculated and the approximated value is then considered newer Output value used in equation (4.3.2).

Using parameter n _d , as in Section 4.2, the course of the activity distribution can then be determined according to the possibilities in Fig. 3.

For t = 0 and x = 0, the initial conditions n = N ₀ 99999 00070 552 001000280000000200012000285919988800040 0002003912995 00004 99880 and s = 0 are present.

If the value χ is neglected once in (4.3.2), the condition n = N ₀ is given at t = 0.
N ₀ is calculated from (4.1.15).

In this numerical solution, the material properties in the cleavage product deposition should be taken into account in the approximation of the solutions already in the expression F.

For the equation system, all approximation methods suitable for the equations of the type dy / dx = f (x, y) can be used.

5. Task of the method in the determination of the cleavage product deposition in industrial application

After the physical basics in section 2 to 4 have been considered, it is possible with the help of Solution in Section 4, a method for determining the To develop fission product deposits.

Thus, the models for the cleavage product deposits in the Technology and reactor safety, it must Possibilities give the necessary parameters for the Specify material properties.

The explanations in section 2.4 show that a provision the parameter from the recalculated activity distributions is not sufficient in the evaluation of the reactor experiments. Thus, the parameters in different reactor experiments can be compared with each other, it is not enough that one in the model calculations at its own discretion decides whether at the match of calculated and measured Activities the parameters are good or bad.

The example of the evaluation according to literature [12] and also in [11] shows that with this method of evaluation a prediction the Spaltproduktablagerung in HTR plants very questionable and seems hopeless.

The developed procedure should make clear that at the difficult problem is the possibility of one Prediction of incoming activity profiles metallic reactor components in HTR systems.

5.1 equations for the procedure

With the solutions (4.1.14) and (4.1.16) it is possible, as in section 3.1, the method of least squares apply.

For the solution (4.1.16) should be a reduced regression line be found so that the parameters for the equation (3.1.16) or for (3.1.21) from the method of the smallest Squares can be determined.

To derive the equation for the reduced compensation straight, the solution (4.1.16) by logarithmation on the  brought the following form

The abbreviations A ₁ and A ₂ have the following shape

Using the abbreviation (4.1.5) and χ = λ , the second term on the right can be transformed into (5.1.1). You get

With

With the expression for F in (4.1.19) and with the replacement of d n _a / d t with d n _n / d t = A - D , one obtains first

Substituting F , in (5.1.4), into the equation (5.1.2) yields the expression

and

Substituting equation (5.1.5) into equation (5.1.1), so you get

From the equation (5.1.7), using other terms for material flow densities A and D , for adsorption and desorption, the reduced least squares regression line can be derived.

How many variables will be contained in the regression line depends on the number of parameters to be investigated in the material flow density A.

The degree of coverage σ , in equation (5.1.7), can be determined by σ = n _a / n _o from equation (3.1.5) with the help of the measurements from the reactor experiments.

With the use of the established equations in the Sections 3 and 4, should be given an example how to the equation for the best-fit line in other cases as well from the expression (5.1.7) can derive.

In order to take account of any additional saturation effect during the adsorption process, equation (3.1.21) is used for the material flow density A , in (5.1.7).

For the material flow density D , in (5.1.7), the expression (3.2.3) is used.

If one sums the elements separately to account for the particle decay in A and D , as in (4.2.1) or (4.2.2), then with additional division by the collision number Z , σ and f ( σ ) one obtains the equation, still has to be converted to a reduced straight line, in shape

All sizes that you can specify yourself and from the measurements are on the left in (5.1.8), the then the value range of the function to be searched for Show balancing line.

All necessary parameters that can be determined for the adsorption process can be found on the right-hand side in (5.1.8), which form the domain of the function to be searched for. Truncate to the left side in (5.1.8) with U = U (σ, T, η, Δ E _d), and the logarithm (5.1.8), we obtain the reduced-fit line for the method of least squares

According to the equation (3.1.18), for (5.1.9) following application of the regression line Abbreviations:

Z = ln U (function value)
X = ln (T / T ₀)
Y = 1 / T

For the balancing line of the form

Z = a ₀ + a ₁ · X + a ₂ · Y (5.1.10),

The following abbreviations result for the constants

a ₀ = ln C
a ₁ = m
a ₂ = - Δ E _a / R.

Again, it should be noted in the selection of solutions that in the fit the constant a ₂ 0, so that the Boltmann factor is 1.

The sign of a ₂ depends on the choice of the parameter η in the equation (5.1.8).

When choosing η , U must be 1 to get real numbers for Z.
The value η can only be 0.

If the conditions for a ₂, U and η are satisfied, one can consider the determined constants C, m and Δ E _a as physically meaningful parameters.

For the selection of the parameter η , one can set another restrictive condition, so that the parameters no longer need to be determined at their discretion.

5.2 Solution of the task in the determination of the parameters by means of the method

The remarks in this section are intended to show that you have the necessary material parameters from the still can determine the following conditions when using the procedure equations (5.1.8) and (5.1.9) are used.

In the evaluation of a reactor experiment can at a variety of the nuclide, the parameters C, m and Δ E _a under the condition of a ₂ 0, as described in section 5.1, from (5.1.9) are determined. In the case of an additional desorption, the desorption energy must be specified as a parameter.

The value of the desorption energy Δ E _d can be estimated from the sublimation energy, if it is assumed that the desorption energy is close to the sublimation of the amount by.

The estimate of Δ E _d is then based on the position of the maximum in the curve 4, to check for Fig. 3.

When the estimated value for Δ E _d, the calculated and the measured activity distribution must the position of the maximum, are at the same surface temperature in Fig. 3, at the same location.

The quantitative agreement of the two curves at the maximum is achieved by the source strength N ₀, by the choice of the gas diffusion coefficient D _G and by the value for n ₀, if the parameter η has been specified.

The quantities which determine the initial concentration N ₀, the mass transfer coefficient H and the value n ₀ from the measurements should be known in advance if possible.

Therefore, it is necessary that one in the experimental design the flow conditions, by choosing a favorable Geometry, as well as possible about known mass transfer laws can capture.

The choice of the parameter η results in a next condition that can be derived for a given degree of corrosion from the statistical properties of the fit. The parameter η is suitably selected if the standard estimation error has a minimum between the values of the fit and the measured values.

The formula for the standard estimate error from Literature / 18 /, which is applied to the equation (5.1.9) and to (5.1.10), has the form due to Z = ln U

Z _gem are the values for Z = ln U , if in U the measured values of the reactor experiment are considered.
Z _Fit are the values for Z = ln U from the regression line, if these are estimated from the equation (5.1.9) or (5.1.10).
N is the number of measurements or here also the number of segments in the experimental setup for which the activities for the individual nuclides have been measured.

For the calculations in Section 7, one has for (5.2.1) one other relationship used.

Assuming that in Equation (5.2.1), when η is chosen, the standard error has reached its minimum, then the error between the measured activities and the estimated activities from solution (4.1.16) has a minimum.

For the formula (5.2.1) one uses in the calculations the equation

A _gem is again the measured activity in Bq / m².
A _L is the calculated activity from equation (4.1.16) in Bq / m².
N is again the number of measured values.

The calculated activity is obtained from (4.1.16) on the basis of the law of decomposition by the equation
A _L = n ₀ · λ · σ (x, t) (5.2.3).

With the condition s _F = min., The equation (5.2.2) defines the parameter η .

To check the condition s _F = min. It is necessary that after the determination of the constants C, m, Δ E _a , from the equation (5.1.9) given η , the course of the activity using the equation (4.1.16) in the equation (5.2. 3) must be checked to calculate the value of the error s _F from (5.2.2).

The required values for n _a , in (5.1.9), can also be calculated from Equation (3.1.5) in the case of an additional desorption, but in the test with the solution (4.1.16) and (5.2.3) For n _a, based on the expression for F in (4.2.1) and (4.2.2), we search for a suitable parameter to find the course of the calculated activities for s _F. In order to find the parameter n _a for the calculation of F for the solution (4.1.16), we first make, as in Section 4.2, the simplification n _d = n _a , so that for F the equation (4.2.2) is additional Desorption can apply.

As stated in the calculations at Ag-110 m, F has a maximum near the maximum at curve 4 in Fig. 3.

The equation for estimating n _a is obtained from the first derivative of the expression for F after T , assuming the derivative is zero.

The first derivative of expression (4.2.2) for T yields the equation

At one extreme point ofF can the derivative  = 0 set and you get to the estimate ofn _a the expression

For T , we have to use the surface temperature in (5.2.5), at which the maximum of the curve 4 , according to Fig. 3, and F is.

The required constants C, m and Δ E _a have already been calculated by the equalizer line in (5.1.9), so that these are available in equation (5.2.5). The 2nd derivative from F to T must then be at a maximum less than zero.

For the second derivative of F to T we obtain the expression

For the surface temperature, the temperature at the position of the maximum of the activity distribution and F is again used.

The assumption for a monolayer n ₀, which is a constant for all models, can change its values under the conditions of reactor experiments, so that it no longer compares with the values in the experiments in literature [13] and [14], because About 260 different fission product nuclides must find their possible sites during the adsorption process on the material surface. In order to be able to check the experimental values of n ₀ in the evaluation of the reactor experiment, it may be required that the values of F should be about 1, or at least close to 1, when applying equation (5.1.9). This condition can be deduced from the equation (4.1.2) if one assumes that the wall concentration n _{w may} only be at most as large as the gas concentration n .

With the constant n ₀, one can additionally adjust the calculated activities to the measured activity distribution, if necessary.

The value for n ₀ and the parameter η are the reason why, as in Section 1.1, one should only compare reactor experiments with measuring times of different lengths if the experiments were carried out with the same degree of corrosion of the materials and with the same hot activities of the high-temperature reactors ,

This would be possible if reactor experiments at HTR plants with about the same thermal power would be performed to approximately, without much effort, the conditions in section 1.1 to reach.

The following points are intended to illustrate the application steps for the Summarize procedure according to equation (5.1.9).

• 1. The procedure is repeated in the specification of η until U is 1. (Real solution for Z = ln U)
• 2. From the values of h for U 1, the constants C, m and Δ E _a from the fit are calculated according to equation (5.1.9). Here, a ₂ 0 must be satisfied in (5.1.9), otherwise the procedure must be repeated for a new default for η with the condition in point 1. For additional desorption, Δ E _d from the sublimation energy shall be estimated and considered in (5.1.9) before the start of the calculation.
• 3. To check the calculated curve, using equation (5.2.3), with the solution (4.1.16), estimate the value of n _a in the expression for F. Here, n _a itself or from the equation (5.2.5) can be calculated.
• 4. With the calculated activity curve, the error s _{F is} calculated in equation (5.2.2), which is used to replace the standard error for the approximation in (5.1.9).
  The value of η , under the specified conditions, is to be approached until s _{F has} a minimum between measured and calculated activities.
• 5. For additional desorption, the position of the maximum in the calculated activities should be shifted to the position of the maximum of the measured activities. (Curve 4 in Fig. 3)
  If the shift of the maximum by the choice of Δ E _d, n _a and n ₀.
  The quantitative agreement of the calculated values with the measured values is obtained by N _o , D _G and n ₀.
  So that a good fit of the fit to the measured values in curve 4 takes place when equation (5.1.9) is used, the measured values which are clearly above the start and end values of the measured activities are to be used only in the vicinity of the maximum.
• 6. If for the activity distribution, according to Fig. 3, the curve 2 or 3 is present, then n ₀ must be adjusted so that F 1 or at least F ≈ 1 and the condition n _w n is satisfied.

If the case F <0 is present for an additional desorption, one has to expect larger values for K and thus (4.1.16), at higher temperatures, smaller degrees of coverage in the curve 4 , according to Fig. 3 (4.1.5) ,

The curve 4 is determined at higher temperatures and in the course to the maximum of the activity distribution of the net rate of the material flow densities of adsorption and desorption processes.

The course from the maximum to smaller temperatures is through determines the influence of the mass transfer.

The advantages of this method with the equation (5.1.9), compared to the previous evaluation methods in section 2.4, lie in the investigation of the parameters for the Material properties.

Would you like in literature [12] only records with the help Try the numerical solution methods, then how in [12], the case that one with an additional Desorption does not find a maximum in the calculated values. Trying out the parameters to find a maximum would be even more time consuming than this method.

Similarly, equation (5.1.9) shows that the Desorption energy for Ag-110 m near the sublimation  energy for silver, as required by [12] and the value is given as 67 Kcal / mol.

Even with the SPATRA program according to literature [11] at Ag-110 m desorption energies, which at about 60 Kcal / mol and because of (2.2.7) there are some restrictions must accept.

This method has been used in Incoloy 800 H in (5.1.9) for silver a desorption energy of 67 Kcal / mol taken into account in order to describe the measured course well.  

5.3 Possibility of predicting occurring activity distributions during fission product deposition in HTR plants

For predictions about occurring fission product deposits on Metallic components in HTR systems, it is necessary that one the behavior of the necessary parameters one from one Reactor experiment according to the conditions in section 5.2 can determine over a longer period depending on knows about the degree of coverage.

Should be used in reactor experiments of different lengths the parameters are no longer in the evaluations as prove constant, then they can only represent functions depending on the degree of coverage and not on the duration of the Measuring time.

Reactor experiments give the same results Hot gas activities for the parameters following substitutions:

If a relation between m ( σ ) and Δ E _a ( σ ) can be found in the form of Eqs. (3.1.19) and (3.1.20) then one can also use f _m ( σ ) in the substitution function from ( σ ) .

To determine the functionality of the replacements, the parameters C are first (s), Δ E _a, m (σ), Δ E _d (σ), n ₀ and η (σ), according to the method in Section 5.2, each of the Reactor experiments at different measuring times and same hot gas activities determined.

After determining the necessary parameters, the values become applied as a function of the degree of coverage to to determine the equations for the functions.

These equations of functions are used in the differential equation system (3.2.10) and (3.2.11) or in the system (4.3.2) and (3.2.11), whereby the functions are also included in the expression (4.3.3). for F and (5.2.5) for the parameter n _a .

If the functions for the parameters are known, then there is the possibility of predicting the distributions of activity from the solutions of the differential equation system (3.2.10) and (3.2.11) or from the solutions of (4.3.2) and (3.2 .11), if the operating time of the reactor is specified until a possible accident occurs. For the flow conditions, z. B. in the steam generators, then must be changed due to other geometric dimensions, as in the experimental arrangements in reactor experiments, the mass transfer coefficient H.

The required mass transfer laws can be partly from the Derive heat transfer laws in literature [23].

As a simplification for the mass transfer law in steam generator one can assume that the mass transfer as in should take place a streamed from the outside pipe.

Also in the dimensions of other metallic components (Hot gas channels) can with simplified assumptions the Mass transfer coefficients from the heat transfer laws in [23].

The determined parameters or their functions, in Depending on the degree of coverage, must be specified in the Differential equations used and out of the reactor be maintained experiments.

The advantage of using this method lies in the Saving costly laboratory experiments, where to get under Reactor conditions also has no absolute certainty that the results of these experiments can be used. If there is the possibility for a comparison of results from laboratory and reactor experiments results, one should on do not give up this.  

6. Compilation of necessary equations in the model calculations

For the calculation of mass transfer coefficient according to formula

H = Sh · D _G / d _H (6.1),

you need the Sherwood number Sh, the gas diffusion coefficient D _G and the hydraulic diameter d _H for the experimental setup. To derive the mass transfer law from the heat transfer law, the Nusselt number is replaced with the Sherwood number, the Prandtl number against the Schmidt number and the thermal conductivity against the gas diffusion coefficient, if the heat transfer coefficient is to be replaced by the mass transfer coefficient.

This can be done with gases, as the heat and heat indicators Mass transfer are the same.

6.1 Calculation of Sherwood Number

From the heat transfer law for a turbulent pipe flow

Nu = 0.037 × (Re ^0.75-180 ) × Pr ^0.42 (6.1.1) [20],

we obtain the equation for the mass transfer law

Sh = 0.037 × (Re ^0.75-180 ) × Sc ^0.42 (6.1.2).

The scope for (6.1.1) is in the range

2300 <Re <2 × 10⁵ and 0.6 <Pr <500.

Nu is the Nusselt number.
Pr is the Prandtl number.
Sh is the Sherwood number.
Sc is the Schmidt number.
Re is the Reynolds number.

The additional factor (1 + (d / L) ^2/3 ) · ( η _F / η _w ) ^0.14 , which would have to be added in (6.1.1), has, as in literature [20], with assumed the value 1.

Here d is the pipe diameter.
L is the pipe length.
η F is the fluid viscosity at medium fluid temperature.
η _w is the fluid toughness at a wall temperature T.

If, according to literature [16], the tube wall temperature is the same as the temperature of the flowing fluid, then the toughnesses are equal and the additional factor can be assumed to be 1.
Here, too, d = d _H « L.

6.2 Calculation of the Schmidt number

The Schmidt number can be derived from the equation

determine.

 is the kinematic tenacity.
 is the dynamic viscosity in N · s / m².
ρ is the density of H_e in kg / m³.

6.3 Calculation of the Reynolds number

The Reynolds number is determined from the formula

6.4 Calculation of the flow velocity

The flow rate is calculated from the expression

6.5 Calculation of the density for helium

The density of helium can be controlled with sufficient accuracy derived from the equation of state for ideal gases. The equation then has the form for the density

m _u is the atomic mass unit in kg.
p _He is the pressure of helium in the primary circuit in N / m².

6.6 Calculation of the dynamic viscosity

The formula obtained from the data of the dynamic viscosity depending on the temperature in the literature [21] has determined is

 = 4.3088 · 10^-7 ·T ^0.67148 Ns / m² (6.6.1).

T is the temperature in K.

6.7 Calculation of the gas diffusion coefficient

For the diffusion of 2 gases, the following formula was used from the literature [15] used

With N ₁ = p ₁ / K _B T and N ₂ = p ₂ / K _B T, one obtains from the equation (6.7.1) for the gas diffusion coefficient the expression

M ₁ is the atomic mass of He in kg.
M ₂ is the atomic mass of the nuclide in kg.
σ ₁₂ is the average joint cross section in m².
p ₁ = p _He is the gas pressure for He in the primary circuit in N / m².
p ₂ = p _d is the partial pressure of the nuclide in N / m².

For p ₂ « p ₁, the partial pressure of the nuclide in equation (6.7.2) can be neglected.

The average joint cross section was with the relationship

σ ₁₂ = 4 π · (r + r ²₁ ²₂) / 2 (6.7.3)

wherein r ₁ and r ₂ are the atomic radii of the helium atom and the nuclide of Table 4.

From the equation (6.7.2) can be found for the nuclides estimate the following gas diffusion coefficients:

Cs-137 D _G = 5.14285 × 10 ^-4 T ^03.02 / p _He m² / s Cs-134 D _G = 5.14449 · 10 ^-4 T ^3/2 / p _He m² / s Co-60 D _G = 9.54399 × 10 ^-4 T ^03.02 / p _He m² / s Fe-59 D _G = 9.54905 × 10 ^-4 T ^03.02 / p _He m² / s Cr-51 D _G = 9.35789 · 10 ^-4 T ^3/2 / p _He m² / s Eu-154 D _G = 4.39145 × 10 ^-4 T ^03.02 / p _He m² / s Eu-155 D _G = 4.39109 × 10 ^-4 T ^03.02 / p _He m² / s Sc-46 D _G = 6.67124 × 10 ^-4 T ^03.02 / p _He m² / s Zn-65 D _G = 8.51488 × 10 ^-4 T ^03.02 / p _He m² / s Ag-110 m D _G = 7.46738 × 10 ^-4 T ^03.02 / p _He m² / s

T is the cooling gas temperature in K.
P _He is the pressure of the cooling gas in the HTR in N / m².

Equations (6.1) through (6.7.3) are all prerequisites given for the calculation of the mass transfer coefficient. One has here for the calculation of the mass transfer coefficient the assumption that used the mass transfer in a pipe should take place because in the experimental setup at VAMPYR II / V01 experiment in ring gap arrangement two different materials for test tube and test rod were incorporated into the experimental setup.

The mass transfer law for an annular gap arrangement could also be applied here, but with different materials There are difficulties in getting the initial concentration or the source strength from the hot gas activity for the accumulating shares on the individual experimental materials can not divide before, because you on reason different interactions of the surface with the  Fission products with different material flow densities must count on the adsorption and desorption processes. A suitable distribution of the initial concentration, Definition of parameters for the fission product deposition is in the experimental setup of the VAMPYR II / V01 experiment, even the other models, not possible.

The different interactions of the materials with The fission products may occur is due to the location of the Maximums in the measured activity distribution of Ag-110 m to detect between Incoloy 800 H and Inconel 617 on different desorption energies are due.

Experimental arrangements would become necessary for the reactor experiments are suitable, where for the different materials several Test tubes or pipe segments are arranged one behind the other, because you get the initial concentration for a pipe or pipe segment from the concentration of previously flowed through with cooling gas Test tube or segments, from the consumption of fission product nuclides in fission product deposition, can calculate. In a trial with one or more Test tubes is a determination of the initial concentration possible because only one type of material for the interaction is responsible for the fission products.

Despite the experimental setup of the VAMPYR II / V01 experiment, the following Fig. 6 for Cs-137 should illustrate the values for the mass transfer coefficients, according to the equations in Section 6, with the results in [22].

Even if in section 6 the laminar fraction of Flow has not taken into account in the mass transfer law, so results in comparison with [22] for the slope of the Mass transfer coefficients as a function of the temperature a good match.

The quantitative deviation by the factor of about 2, is on the estimation of the impact cross section in (6.7.3) due.  

6.8 Temperature curve used in the inner tube wall of the experimental setup

For the temperature history you have from the temperature data according to [22] the following equation is determined

T (x) = -111.9649 · x + 1083.3787 T in K (6.8.1).

x is the location in the flow direction in the experimental setup for which the temperature on the pipe inner wall is to be determined. The deviations for the determined temperatures compared to the measurements are less than 2% in equation (6.8.1).

For x , the relationship arises

x = segment no. · Δ x (6.8.2)

With

Δ x = L / number of segments
L is the pipe length in m.

When temperature drops in piping, you can look Literature [16] the temperature profile from the formula

to calculate.

T₁ is the inlet temperature atx = 0.
T₂ is the exit temperature atx =L,
T _a is the outside temperature of the pipe.
Φ _LO  is the heat loss, atx = 0 withT _i =T₁, in W / m.
T _i is the internal temperature atx = 0.
c _p is the isobaric specific heat capacity of the fluid in KJ / kg · K.
 is the mass flow in kg / s.

If (T ₁ - T ₂) « (T ₁ - T _a ), then one can linearize (6.8.3) and obtain the equation

A linear temperature curve in (6.8.4) is also given approximately in (6.8.1).  

6.9 Estimation of the gradient for the flow velocity

The gradient for the flow velocity of the cooling gas can be estimated from the density change of He.

If we substitute equation (6.5.1) into equation (6.4.1) and then differentiate the expression obtained for the velocity after T , one obtains first

From the equation (6.8.1) we obtain the expression by differentiation to x

Using the equations (6.9.1) and (6.9.2), then yields for the gradient of flow velocity

For ∂ T / ∂ x one can derive the following expressions from equations (6.8.3) and (6.8.4)

In (6.9.10), in the derivative ∂ T / ∂ x the temperature T ₂ is considered in equation (6.8.3), which is derived from x . For L , place x where the velocity gradient should be considered.

The velocity gradient, in (6.9.3), has due the heat transfer from the fluid to the pipe inner wall in the experimental arrangement a negative sign.  

6.10 equations for the best-fit line (5.1.10) using equation (5.1.9)

For the balancing straight

Z = a ₀ + a ₁ · X + a ₂ · Y (6.10.1) (5.1.10)

If you have the following system of equations to solve the constants:

Σ Z = a ₀ · N + a ₁ · Σ X + a ₂ · Σ Y

Σ XZ = a ₀ · Σ X + a ₁ · Σ X ² + a ₂ · Σ XY (6.10.2)

Σ YZ = a ₀ · Σ Y + a ₁ · Σ XY + a ₂ · Σ Y ²

The partial correlation coefficients are calculated following equations:

The multiple correlation coefficient is calculated the expression

N is the number of measured values.

7. Exemplary embodiments and discussion of the results

The values in Tables 13 to 29 are the results that from equation (5.1.8) via equation (5.1.9) have been.

For the calculations for Ag-110 m, the activity distributions were calculated once using equation (4.2.1) and additionally using equation (4.2.2) in (5.1.9). In equation (5.1.8) one has only to set n _a = n _d to take the case into account in equation (4.2.2).

Assuming that the data for the determined hot gas activities are correct, the case n _a = n _d in the equation (4.2.2) gives a good agreement with the measurements, if one expects the desorption energy in the vicinity of the sublimation energy of silver.

In applying equation (5.1.9), only the measured values which contribute substantially to the activity in the environment from the maximum have been used to determine the parameters C, m and Δ E _a . In the case of the Incoloy 800 H, the values with the segment numbers 16 to 21 in Fig. 7 and 9 have been taken into account for Ag-110 m. For the system of equations of the equalization line (5.1.9), the number of measured values is therefore N = 6.

For Inconel 617 the values with the segment numbers 18 to 27 were used and the number of measured values is N = 10. The calculations for Ag-110 m show a good agreement with the results from the literature [11].

Also in literature [11] one has the values of gas diffusion must change coefficients, thus the course of the maximum to the exit temperature with the measurements well matches.

Because of the still uncertain values of n ₀ and the estimated values for the mass transfer coefficient, these calculations can not be considered as the final definition of the parameters but as a way to see that the method is suitable for fulfilling the conditions To determine the parameters so that the reactor experiments at different measuring times can be compared with each other, when the hot gas activities in the experiments are approximately equal.

In all calculations f ( σ ) = 1 has been assumed in equation (5.1.8) because the degrees of coverage are very small. In the calculated activity distributions of Ag-110 m, it has not necessarily been ensured that in equation (5.2.2) the value for s _{F has} a minimum.

For the accuracy of the parameters, the agreement was calculated values in the vicinity of the maximum at the Measurements crucial to the gas diffusion coefficient for to estimate the mass transfer coefficient.

The tables from the literature [11] and the illustrations from the literature [12] are listed, so a comparison different results is possible.

For the other nuclides, the value for s _{F has} a minimum, so that the parameter η is fixed.

The attached figure from the literature [10] also shows for the experiment sapphire 05 similar courses of activity distributions, as in the experiment VAMPYR II / V01.

The course of J-131, the steel tube (4541) from Fig. 27 of Literature [10], shows that similarities with the course of Zn-65 in Incoloy 800 H exist, so that the possibility exists, the method on non-metallic To use fission products.

The flat course in most nuclides could be due to Cs-137 and Cs-134 are caused to be affected by Cs ions to a development of electrostatic forces on the Material surface could come and therefore the desorption energies are no longer sufficient to allow the fission products can hardly or not leave the surface again. An explanation of the formation of Cs ions on surfaces can be found in literature [24].

This still unresolved cause would be the assumption in literature [1] confirm that Cs predominantly at high temperatures elemental on the material surface is present, so that the assumed dust on the surface not for the flat Course of activity distributions would be responsible. The lack of desorption would be explained by the fact that the necessary desorption energies are smaller than that Coulombic energies when Cs ions electrostatic bond  should cause forces on the material surface. The influence of chemical processes would be in equation (3.1.16) and (3.1.21).

In Table 29, the partial and multiple correlation coefficients for the individual calculations for the equation (5.1.9) are summarized to show that an additional temperature behavior in the form (T / T ₀) ^m in Eq. (3.1.16) and (3.1.21) when considering the substance transfer.

For the arbitrarily chosen reference temperature T ₀, one can also use the characteristic temperature in literature [16] for the thermal conductivity of the material. Another arbitrary reference temperature T ₀ provides other values for the parameters C and m .

In applying equation (5.1.9), the equation System (6.10.2) solved with the Gauss-Jordan method to to avoid numerical problems.

If only the Gaussian algorithm is used, then the values for the parameter m are too high and one has to increase the number of measured values N in the equation (6.10.2) until the values for m decrease dramatically.

For the calculations in Tables 13 to 29, the values are for the number of measurements for equation (6.10.2), the numerical problems also in the Gauss-Jordan method be able to fix.

The specified values for N at the end of the tables differ from the actual number of measured values.

By using computers with double precision in the programming languages, the numerical Solve problems.

The fluctuations in the measured values are probably on the Attributed degree of corrosion.

Information on the degrees of corrosion, for the individual segments in the experimental design, were not available.

The pressure losses in the annular gap arrangement were at VAMPYR II / V01 experiment neglected.  

Temperature curves in the annular gap arrangement in Experiment VAMPYR II / V01 from the literature [11].

Table 9

Table 10

Values for measured and calculated plate-out profile for Ag-110 m for Incoloy 800 H

Table 11

Values for measured and calculated plate-out profile for Ag-110 m at Inconel 617

Table 12

Determined desorption energies, gas diffusion coefficients as well as swelling strengths and roughness factors used in the evaluation of the deposition profiles in silver

Table 13

Table 14

Table 15

Table 16

Table 17

Table 18

Table 19

Table 20

Table 21

Table 22

Table 23

Table 24

Table 25

Table 26

Table 27

Table 28

Table 29

Table 29

Information from the basis of calculation from the literature [12]

2. Basis of calculation Test track geometry according to [1] operating conditions He mass flow | 2.28 g / s print 10.6 bar uptime 54 days Swelling rate at the entrance of the test track @ Cs-137 7.3 x 10⁹ atoms / s Ag-110m 5.6 x 10⁷ atoms / s gas temperatures @ Input Output 750/357 ° C Pipe wall temp. (Table 1 [1]) @ Input Output 725/400 ° C

materials pipe Incoloy 800H Rod Inconel 617

Details of the parameters used for the results in literature [12]

Deposition parameters for Cs-137 and Ag-110m

8. Summary

As explained in Section 5, the procedure for Determination of the necessary material parameters, for determination the adhesion coefficients of the metals and metal alloys Components in the primary circuit of HTR systems, at the Evaluation of reactor experiments can be used with Help of an analytical or numerical solution predictions about occurring activity distributions in HTR systems up to to make the possible occurrence of a fault.

The concept from which the method was developed is based on the equations (5.1.5), (5.1.4) and (5.1.1).

When considering other interactions of the Fission product nuclides with the material surface, is the Net rate of mass densities for adsorption and desorption

d n _n / d t = (A - D) = Hn _w

replaced by other expressions, so that the user from the equations (5.1.5), (5.1.4) and (5.1.1) another analysis method for the evaluation of reactor experiments can develop.

The necessary additional equations can be found on p. 54. In Section 6, the equations used serve as additional overview needed by the user and through other expressions can be replaced.

The method for determining the material parameters is based on the equation (5.1.7), which consists of the equations (5.1.5), (5.1.4) and (5.1.1), with the corresponding additional equations on p. 54, was derived from the concept.

If the material flow density D for desorption according to equation (3.2.3) is maintained, no new method needs to be developed when investigating the material parameters in the adsorption current density A.

The number of variables, in the least-squares regression line, depends on the number of parameters in the expression for A.

The steps for the determination of material parameters and Determination of the activity distributions in the fission product deposits see section 5.2 on p. 57-61.  

When considering the surface roughness by the parameter η , the method requires solutions (4.1.16) and (4.1.14) with the associated additional equations in section 4.2.

It can also be a numerical solution, z. B. as in Section 4.3.

The procedure offers the possibility that one at the Evaluation of reactor experiments of time and costly laboratory experiments in determining the Material parameter is independent.

The user can, according to the concept of determining the Split product deposit, its own software programs so you do not use any other software programs must take over.

On the described method and concept are based on the technical application claims levied.

The equations used to describe the Adsorption current density in section 3 has one of the Evaluations of the experiment VAMPYR II / V01 received, under the assumption that in the Spaltproduktablagerung no involved in chemical processes.

The equations (3.1.16) and (3.1.21) apply to all In the VAMPYR II experiment, nuclides were considered to be so no special cases to describe the adsorption process needed.

Even if the activation energy Δ E _a ≈0 for many nuclides, the consideration of the activation energy serves to determine the Boltzmann factor e ^{- Δ E} a / ^RT 1, so that the temperature behavior ( T / T ₀) ^m in the equation (3.1. 16) correctly determined.

literature

[1] HJ Fett, J. Herion, EA Niekisch
Cs deposition on HTR materials and thermochemistry of Cs in the HTR refrigeration cycle
KFA Jülich, reactor conference Düsseldorf, p. 107-109
[2] TS Kress, FH Neill
Calculating Convective Transport And Deposition Of Fission Products
September 1968, Contract. No. W-7405-eng-26, pp. 3-7
[3] N. Iniotakis
Current state of theory and file in the field of plate-out at KFA / IRB
Internal Report, pp. 1-8
[4] GerdWedler
adsorption
An introduction to physisorption and chemisorption
Verlag Chemie 1970, pp. 23-24, p. 41-42
[5] Ch. Whitecoat, C. Haman
Basics of solid state physics
Springer-Verlag, Berlin-Heidelberg-New York, 1979, pp. 224-228
[6] F. Reif, Fundamentals of Physical Statistics and the Physics of Heat
Walter de Gruyter, Berlin, New York, 1976, p. 315, p. 320, p. 321
[7] TS Kress and Paul Nelson, Jr.
Numerical Solution of the Isothermal Fission Product Deposition Equation: The Program Predep II, USA
Report ORNL-TM-1970, Oak Ridge National Laboratory, October 1967
[8] K.-D. Ehrhardt
PLATO, a program for the calculation of fission product deposition
Jül.-Spez.-135, Dec. 1981, p. 2-5
[9] HG Honest
Manual for the program SPATRA
[10] N. Iniotakis, KH Münchow
Theoretical evaluation and interpretation of deposition studies in the reactor experiments vampyr / AVR and sapphire / P'gase
Reactor Meeting (1975), Nuremberg, p. 792-795
[11] E. Frank
Investigation of fission product deposition on surfaces of metals
Internal report at KFA / ISF
[12] Röllig, dr. Christian
Preliminary evaluation of the experiment VAMPYR II / V01
High temperature reactor construction GmbH HRB
Mannheim, March 28, 1988, pp. 1-4
[13] RG Wilson, J. Appl. Phys. 37, 3161 (1966), p. 3161
[14] JB Taylor, I. Langmuir, Phys. Rev. 44, 423 (1933), p. 425
[15] Brockhaus, physics
VEB FA Brockhaus Verlag Leipzig, 1972, p. 145, p. 610, p. 250
[16] U. Grigull, H. Sandner
heat conduction
Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1986, pp. 22-23
[17] Charles E. Mortimer
chemistry
The basic knowledge of chemistry in key areas
Georg Thieme Verlag Stuttgart, 1976, p. 56
[18] Murray R. Spiegel, PhD
statistics
McGraw-Hill Book Company GmbH, Schaums Outline, pp. 269-272
[19] FH Neill
Adsorption And Desorption Of Iodine On Mild Steel
ORNL-TM-2763, 1970, p. 33-34
[20] F. Stelzer
Heat transfer and flow
Verlag Karl Thiemig KG Munich, Vol. 18, pp. 139-140, p. 147
[21] Dieter Smidt
Reactor Technology Vol. 1
Verlag G. Braun, Karlsruhe, p. 165
[22] scribe
Private message
Interatom, Bergisch-Gladbach
[23] VDI Heat Atlas, 3rd Edition (1977), Chapter G
[24] G. Herrion
The influence of wall materials on the deposition of Cs in HTR cooling circuits
Jül-1155, January 1975, p. 31
[25] Biedermann et al .:
"Gamma spectroscopic follow-up of the experiment VAMPYR II / V01"
IRB-05-BI, TN-10-87 of October 5, 1987

Designations of physical quantities and abbreviations in the individual sections Designations in section 2.1

N _i is the density for the varietyi on fission products in kg / m³.
N _wi is the density for the varietyi at fission products on the Material surface in kg / m³.
A is the cross-sectional area of the experimental set-up in m².
P is the circumference for the cross-sectional area in m.
 is the mean flow velocity in m / s.
H is the mass transfer coefficient in m / s.
x is the location in the flow direction in m.
λ _i  is the radioactive decay constant for the speciesi  on fission products in 1 / s.
 is the incoming mass flow into the experimental setup in kg / s.
_out is the exiting mass flow in kg / s.
_St is the mass flow through mass transfer to the Material wall in kg / s.
r _a = dn _a/ dt is the mass flow density for the adsorption process or the adsorption rate in atoms / m².s.
n _a =n is the number of occupied detention places on the surface at the timet in atoms / m².
n₀ =n _m is the number of possible detention places for the Coverage of the wall surface at a monolayer in atoms / m².
M _s is the mass per area for a monolayer in kg / m².
M is the mass per area for the number the occupied custody places with a sorti on fission products on the material surface in kg / m².
N _w =N _wi is the density for the wall concentration Fission products in the equation (2.1.12) from the literature [2].
K is a constant in equation (2.1.12) from literature [2].
N₀ is the density atx = 0 for the concentration of Fission products in one varietyi in kg / m³.
R =n _a/n₀ is the degree of coverage.
χ (R) is the condensation coefficient in literature [4].
is the vapor pressure for the varietyi on fission products over the material surface according to equation (2.1.13) in N / m².
 p is the partial pressure for a varietyi on fission products in N / m².
Z is the number of impacts in atoms / m² · s.
s is the likelihood
m₀ is the mass of the nuclide in kg.
K _B = 1.380664 · 10^-23 Ws / grd is the Boltzmann constant.
T is the gas temperature or the temperature the wall surface in K.
R = 1.9872 x 10^-3 Kcal / K · mol is the general gas constant.
Δ e _a is the activation energy in kcal / mol.
Δ e _d is the desorption energy in kcal / mol.
Δ H is the heat of condensation in Kcal / mol.
K₁ is a pressure constant of the equation (2.1.13).
δ (R) is the desorption coefficient in atoms / m² · s.
f '( R ) is a general term for an occupancy-dependent Function in the description of desorption processes in the literature [4].

Designations in section 2.2

x is the location in the flow direction in m.
r is the mean velocity of the cooling gas in m / s.
n (x, t) is the concentration for the varietyi on fission products in the cooling gas flow in atoms / m³.
n _w (x, t) is the wall concentration for the speciesi on Fission products in atoms / m².
H = ShD _G/d _H is the mass transfer coefficient in m / s.
Sh is the Sherwood number.
d _H is the hydraulic diameter of the experimental setup in m.
D _G is the gas diffusion coefficient in m² / s.
V is the gas volume of the cooling gas per unit length in m².
O is the geometric surface of the flow channel per unit length in m.
λ =λ _i  is the decay constant in 1 / s.
Λ is the mean free path in m.
is the desorption current density in equation (2.2.2) in atoms / m² · s.
A is the adsorption current density in equation (2.2.2) in atoms / m² · s.
A is the atomic mass in equation (2.2.15).
 D is the temporal change of the degree of coverage by Desorption in 1 / s in equation (2.2.9).
 is the temporal change of the degree of coverage by Adsorption in 1 / s in equation (2.2.9).
M₀ is the number of possible sites on the material surface, according to literature [8], in atoms / m².
M₀ is the name in equation (2.2.5) according to literature [3].
M₀_∞ =M₀ is the possible number of detention places for one Monolayer, according to literature [3], in atoms / m².
a is the sticking coefficient.
v _⟂ is the impact velocity of the particles or the Nuclides on the wall surface in m / s.
 =v _⟂·a is an abbreviation in equation (2.2.8).
 is the roughness factor according to literature [3].
β is the adsorption coefficient.
(1-β) is the penetration coefficient according to literature [3].
Q₀ is the sorption energy in kcal / mol.
Q _s is the sublimation energy in Kcal / mol.
θ (σ) is a coverage wheel dependent desorption constant, according to equation (2.2.6), in 1 / s.
θ₀ =ν is the frequency of the lattice vibrations in 1 / s.
Q ( σ) is the desorption energy, in equation (2.2.7), in kcal / mol.
σ =n _a/n₀ is the degree of coverage.
μ is the abbreviation in equation (2.2.15).
 is the abbreviation in equation (2.2.16).
D is the material diffusion constant in equation (2.2.18) according to literature [3].
I _D Diffusion current for diffusion into the wall material.
I _e Electricity for the incident particles (nuclides) the material wall.
I _R Electricity for the particles (nuclides) coming from the material wall be reflected.
M is the particle mass in equation (2.2.19).
Φ (x, r, t) is the fission product amount that is in the material diffused.
r is the radial one assuming cylinder geometry Distance.
R _i is the inner radius.
R _a is the outer radius.  

Designations in section 3.1

J is the number of atoms at a timet per unit area depart from the surface, in atoms / m² · s.
J↓ =r _a is the mass flow density for adsorption in Atoms / m · s.
f ( σ ) is a function for description in the adsorption, the may depend on the structure of the solid-state lattice u 03912 00070 552 001000280000000200012000285910380100040 0002003912995 00004 03793nd is described by equation (2.1.6).
p _d =p is the partial pressure for the nuclides from the Hot gas activity in N / m².
x _A is the measured hot gas activity in FIG. 10^-12 Ci / m³.
p _He is the half-life for the nuclides in s.
τ _2.11 is the half-life for the nuclides in s.
a₀ is a constant in the equations for the required fits.
t _B is the operating time for the experiment in s.
α₁ are abbreviations in the equations (3.1.7) and (3.1.8).
is the name of an assumed adhesive coefficient in the equation (3.1.15).
 =v _⟂ is the impact velocity of the particles on the Wall surface in m / s.
Φ₀ is the mass flow density in equation (3.1.13) in atoms / m² · s.
Φ _a  is the adsorption current density in equation (3.1.14).
C is a parameter in Equations (3.1.16) and (3.1.21), which the condensation coefficientχ (R) equivalent.
m is the exponent of the temperature behavior (T / T₀) ^m  in of equations (3.1.16) and (3.1.21).

All other names are from the previous ones Sections taken over.

Designations in section 3.2

A is the abbreviation for the adsorption current density according to equation (3.2.2) in atoms / m² · s.
D is the abbreviation for the desorption current density in equation (3.2.3).
n _d is an assumed parameter for the number of atoms leaving the surface during desorption at time t , in atoms / m².
η is a parameter for taking into account the surface roughness of the mass transfer.
C ₀ is a constant for C in equations (3.2.13) to (3.2.15).
T ₀ = 273.15 is an arbitrary reference temperature in K.

All other names are from the other sections accepted.

Designations in section 4.1

F is the expression (4.2.1) and (4.2.2) to account for the interaction between the nuclides in the gas phase and on the surface.
N ₀ is the initial concentration or source strength at x = 0 in atoms / m³.
K is an abbreviation for the expression (4.1.5) in 1 / s.
T ₀ Constant in equation (4.1.12) for the time component of the concentration n (x, t) in atoms / m³.
T in equation (4.1.12) is the time fraction for the concentration n (x, t) in atoms / m³.

Otherwise, all other names are taken over again.

w is the probability of leaving the surface of a particle in 1 / s (designation in section 4.2).

Designations in section 5.1

A₁,A₂ are abbreviations in the equation (5.1.1).
G is the abbreviation in equation (5.1.3).
W is the abbreviation in equation (5.1.6).
 is the derivative ofF after the temperatureT.
 is the 2nd derivative ofF after the temperatureT.
s _F Abbreviation in equation (5.2.2).
A _L is the calculated activity in equation (5.2.3) in Bq / m².

Designations in section 6 are needed there Equations listed separately.

Claims (3)

A method of predicting and determining fission product deposits on metallic components in HTR plants and reactor experiments to measure occurring activity distributions on surfaces of metals, characterized in that there is a concept for describing fission product deposits based on a physical model for describing physisorption processes; from which this method was developed to be independent from the use of other computer software. 2. Method and concept according to claim 1, characterized in that it as Analysis procedure for reactor experiments for the measurement of Cleavage product deposits can be seen, if necessary also of additional time and cost consuming Laboratory experiments in the measurement of material parameters to be independent. 3. The method according to claim 1 and 2, characterized in that from the Analysis of Reactor Experiments Activities on metallic components for evaluation of possible incidents and related Downtime during the maintenance of HTR systems be enabled.