During the design of a particular laser system, the need may arise for the prelim­inary estimation and optimization of the spatial distribution of gas density in the laser channel’s active region. The behavior of a gas in systems where the density deviations are comparable to, or exceed, their average values over the channel’s volume can only be correctly calculated using two-dimensional numerical pro­grams similar to that of the Idaho National Engineering Laboratory [44]. In ref. [45], a simplified technique was proposed for the direct numerical calculation of spatial inhomogeneities in the channel of a gas-flowing NPL. The only example of a similar calculation available at the time of publication performed using a simplified technique for determining the specific energy deposition of fission fragments in a channel of finite width with a plane-parallel uranium layer arrangement was also presented in ref. [45] by way of an illustration. A comparison demonstrated that the
difference in the results of density distribution calculations using the procedure described in ref. [45] and the calculations performed using the two-dimensional gasdynamic program described in ref. [44] does not exceed ~2 %.

The fundamentals of the procedure described below [46] consist of the follow­ing. As a gas flows through a plane-parallel channel, the length of which, b, is comparable to its transverse dimension, d, the pressure in the laser channel’s transverse cross-section is almost homogeneous (see the first section of this chapter). The typical gas temperature equalization time in the channel comes to tt ~ d /fo, where a is the thermal diffusivity coefficient of the gas. For the gas mixtures used in NPLs, a < 1 cm2/s; hence, it follows that, at d ~ 1 cm, the temperature equalization time in the channel is tt > 1 s. The time during which a portion of the gas traverses the channel in the x direction is determined by the obvious correlation т ~ b/U. At gas-flow velocities of U > 1 m/s, the average time that the portion of gas is present in the channel comes to т < 0.1 s. We will now apply the technique under consideration for the two-dimensional calculation of the gas flow in a channel to the gas flow velocities at which the time that the portion of gas is present in the channel, т, is much shorter than the typical heat exchange time, tt. In this instance, the heat exchange processes in the bulk of the gas volume can be ignored during approximate calcula­tions. The only exception consists of the thin near-wall passive zone layers, the method for considering the effect of which will be discussed next.

In the first approximation of the proposed technique, the flow lines, and accord­ingly, the gas jets separated along each line are assumed to be rectilinear, while the spatial distribution of fragment energy deposition is calculated under the assump­tion that the gas density distribution is homogeneous throughout the channel volume. The dependence of gas temperature upon the x coordinate is then calcu­lated for each flow line, by means of which, using a state equation and proceeding on the basis of the condition that the gas pressure is almost homogeneous through­out the entire channel, the spatial gas density distribution is calculated. Thereafter, pursuant to the results obtained in refs. [44, 50], according to which the gas’s longitudinal velocity profile, Un(x), is transversely homogeneous (with the excep­tion of the narrow near-wall viscous boundary layer), and based on the requirements of mass flow constancy both within the channel as a whole and in each of the jets separated, the gas velocity and the variation in the transverse dimensions of the jets as a function of the x coordinate are calculated. The latter makes it possible to determine the curvature of the flow lines, and using them to correct the spatial distribution of gas density. During the next step, the specific energy deposition distributions of the fragments are calculated based on the density distribution obtained during the previous step, then the procedure for finding the density distribution is repeated using the arrangement described above, etc.

When determining the basic computational correlations of the stationary prob­lem, it was assumed that the gas’s density, velocity, and temperature distributions, as well as the specific intensity of uranium nuclear fission, are not dependent upon the z coordinate directed along the system’s optical axis (in Fig. 9.1, the z axis is perpendicular to the figure’s plane and coincides with the direction of the laser’s optical axis), i. e., a purely two-dimensional problem was examined. The gas flow in
the planar channel was represented by superimposing the independent jets, the camber and thickness of which can vary with the distance traversed, x. The effect of the viscosity within the jets and the friction between adjacent jets was ignored. Each jet was tagged with an Eulerian coordinate at the inlet, y0, and its width was assumed to be so small that the density, velocity, and specific energy deposition within the confines of any of its transverse cross-sections could be regarded as homogeneous. Heat exchange between the jets was ignored. Then in the active region outside the confines of the passive zone by virtue of pressure homogeneity for a given jet, from the energy balance equation, we get

U(x yo)p(x, yo)T(x, У0)ср5у(Х; Уо) — UoPoToCp8у0

where T(x, y0) is the gas temperature in the jet under consideration at a point with a coordinate of x; U(x, У0) is the gas velocity in the jet under consideration at a point with a coordinate of x; U0 is the gas velocity at the channel inlet; w(x, y0) is the specific power deposition at a point with a coordinate of x for a jet marked with a coordinate of y0; Sy(x, y0) is the thickness of the jet under consideration at a point with a coordinate of x; £y0 is the thickness of the jet under consideration at the channel inlet; p(x, y0) is the gas density in the jet at a point with a coordinate of x; and p0 is the gas density at the channel inlet.

From the condition of mass flow constancy in each jet, we get

U(x, y0)p(x, y0)<5y(x, У0) = ^Po^ (9-143)

then, according to the previous equation,

In the main body of the flow (outside the confines of the viscous boundary layers), the gas’s longitudinal velocity profile, as previously mentioned, is transversely homogeneous (U(x, y0) = Un(x)) and Eq. (9.144) is simplified

For rare gases, the relationship between density, temperature, and pressure is described with good accuracy by the ideal gas state equation

P = cv(r — 1)pT.

Thus, for the gas density in each jet, we get

P 1

— ——— dx

Cp U(x, y-)p(X; Уо)

0

According to Eq. (9.143), the relationship between the y0 coordinate and the y Eulerian coordinate takes the form

Уо 0

The dependence of the near-wall passive zone’s transverse dimension upon time is described by an equation of the (9.142) type. Direct calculations (see Chap. 8) revealed that the proportionality factor for sealed lasers is A « 3 and is slightly dependent upon cell dimensions, neutron pulse shape and intensity, and uranium layer thickness, as well as upon the gas mixture’s thermophysical properties. A passive zone is also formed in the channels of NPLs with gas pumping, the wall temperature of which is lower than the gas flow temperature. Its size is a function of many parameters that reflect the geometry and physical conditions of a flow, especially including viscosity and heat conduction. In the general case, it is not possible to derive an explicit analytical expression for the dependence of passive zone size upon the longitudinal x coordinate. However, since the time required for a portion of the gas to traverse a distance of x in the main section of the channel (outside the viscous layer) is. X

dx

UM

0

it is then proposed that the transverse dimension of the near-wall passive zone at this distance be roughly determined using the correlation

As was shown in the preceding section (also see ref. [51]), even in the simplest case of Un(x) = U0 and w/p = const, the A factor is dependent upon the Prandtl number, Pr = v/a, and may lie within limits of 1.4 < A < 1.9.

During the calculation procedure used in ref. [46], temperature distribution in the passive zone was described by the parabolic approximation

Tp{x, y) = a0 (x) + ai (x)y + a2(x)y2.

The a, coefficients are determined from the conditions at the passive zone boundary and at the channel wall. So, for example, when a constant wall temperature is present, the following conditions are used:

Tp(x, la) = T(x, la); dTp(x, la)/dy = 0; Tp(x, 0) = To.

If the velocity in the flow core is also not dependent upon the longitudinal x coordinate (i. e., Un = U0) then the thickness of the viscous layer is determined by the correlation [2]

lv(x)= . (9.147)

The situation is more complicated when Un is a function of the longitudinal x coordinate. Extant literature source contains no direct recommendations on how to proceed in this instance. Therefore, since the time required for a portion of the gas to traverse a distance of x within the channel outside the viscous layer is determined by Eq. (9.145), it is proposed that the transverse dimension of the viscous layer at this distance be approximately determined using the correlation

The following approximation was used to describe velocity distribution in a viscous layer at y0 < lv(x)

In this case, according to the impulse theorem, R = 4.64 [2]. Such a distribution duplicates the true velocity distribution [2] near a planar surface washed around by an incompressible fluid that has a constant viscosity and a constant velocity outside the confines of the viscous layer with an error of less than 1 %.

Thus, an artificial technique was used to describe the width of the passive zone and the viscous layer, as well as temperature and velocity distribution within their confines. Nonetheless, at sufficiently high velocities, U0, the dimensions of these structures are small; therefore, it was anticipated that the proposed approach would not lead to noticeable errors during the calculation of density, temperature, and velocity distributions in the active zone.

The spatial distribution of the specific power deposition due to fission fragment slowing-down was calculated using the procedure described in ref. [49].

During the third iteration step, the gas density distribution differs from the distribution obtained at the end of the first step by not more than 1 %, which is indicative of the good precision of the iterative scheme used.

A comparison demonstrated that the difference between the density distribution calculation results obtained using the procedure described in ref. [46] and similar calculations [44] performed using a two-dimensional gasdynamic program like the one employed during the calculations in ref. [45] based on a simplified procedure for determining the specific energy deposition of fission fragments, does not exceed

2 %. In addition, it was learned that the variation of the A factor value from 1.4 to

3 only results in perceptible density changes during calculations of the same problem alternative in the passive region; for the active region, these changes do not exceed ~1 %. At a gas-flow velocity of U0 = 4.5 m/s, A factor variability leads to relative fluctuations in the active region volume of ~10 %. These fluctuations diminish with an increase in gas-flow velocity; so, at U0 = 9 m/s, they come to ~6 %. Nonetheless, at all A values, the nature of the behavior of all the dependences cited in this work is preserved.

The channel studied during the calculations, which has dimensions of d = 2 cm and b = 6 cm, has open layers (without a protective film) that consist of metallic uranium with a thickness of <5u = 2 .78 ^m (a normalized thickness of D1 = 0.5). It was assumed that uranium fission intensity in the active layers comes to q = 2 x 1016 cm~3 x s_1. The spatial distributions of density and specific energy deposition in active volume of the gas were calculated for a wide range of helium and argon pressure and velocity variations. It was shown that in the central regions of the gas volume (at distances from the channel inlet and outlet on the order of the third fission fragment range in the gas), the dependence of these parameters upon the longitudinal coordinate parallel to the gas flow direction is almost linear in nature, while their dependence upon the transverse coordinate is close to parabolic. The results cited below correspond to a value of A = 3 (apparently the least probable value [51]), which corresponds to the smallest active region volume.

An important parameter that characterizes a real laser channel with a finite dimension of b along the gas flow direction is the efficiency, e, of total energy deposition in a laser-active region not involved in near-wall passive zones. It is defined as the ratio of the total fission-fragment energy deposition in the active region solely to the total fragment energy released in the uranium layers. It is useful to compare this parameter to the efficiency of energy deposition, e0, in an unperturbed gas in a channel without gas pumping (U0 = 0) that has a homogeneous density distribution (there are no near-wall passive zones). Below, the former will be called the real energy deposition efficiency, while the latter will be called the ideal energy deposition efficiency of an unperturbed gas.

In ref. [37] (also see Sect. 7.5), an energy deposition optimization parameter was introduced for NPLs without gas pumping, excited by fairly short neutron pulses with a duration in the millisecond range,

«0 = Dieo-

where D1 = SU/R1 is the normalized uranium layer thickness; R1 is the range of an average fission fragment in this layer; w0 is the specific power deposition in the center of the laser channel; and (w) is the average value of the specific power deposition over the channel cross-section.

In laser channels with gas pumping that operate in the stationary mode, a near­wall passive zone can reach appreciable dimensions at the channel outlet in the presence of relatively low gas-flow velocities. In addition, gas density steadily decreases in the longitudinal direction. For these reasons, real energy deposition efficiency, e, can differ conspicuously from unperturbed channel energy deposition efficiency, e0; thus, a more rigorous statement of the optimization parameter should be introduced:

where

b

w (x, 0) dx,

while w(b/2, 0) is the specific energy deposition value at a point that corresponds to the geometric center of the channel.

The «1 and «2 parameters are distinguished from «0 by the fact that the calculated real energy deposition efficiency, e, is used therein instead of unperturbed channel energy deposition efficiency, e0. In the first of these, the fraction characterizes the ratio of the specific energy deposition averaged over the channel’s entire central plane, y = 0, to the specific energy deposition averaged over the channel’s cross-section. In the second, it characterizes the ratio of specific energy deposition in the center of the channel to the specific energy deposition averaged over the channel’s cross-section.

The dependences of the «1 and «2 optimization parameters upon argon pressure in the channel at U0 = 4.5 m/s are shown in Fig. 9.22. Here, too, a similar

dependence for the ideal parameter, rn0 (at U0 = 0), is shown. The calculated dependences of the optimum (as far as energy deposition) argon and helium pressures upon gas-flow velocity are presented in Fig. 9.23. We note that the optimum argon pressure in a channel with gas pumping over the velocity interval studied, U0, may exceed the optimum pressure for a sealed unpumped laser operating in a comparatively short (<10 ms) pulse lasing mode (P « 0.6 atm) by

1.5 times (up to P = 0.9 atm), whereas the optimum helium pressure in a channel with gas pumping differs little (not more than 15 %) from the optimum pressure for a sealed unpumped laser. The spatial distributions of argon density at an optimum pressure of P = 0.9 atm and a velocity of U0 = 4.5 m/s are presented in Figs. 9.24 and 9.25 for T0 = 293°K.

The calculations made it possible to find out a quite important fact. At the optimum pressure precisely, the distribution of specific energy deposition in the central plane, y = 0, and in the adjacent planes, y = const, is symmetrized relative to the central cross-section, x = b/2 (see Fig. 9.26b), equalizing in a considerable

portion of the gas volume. At pressures lower that the optimum value in a gas region that stands away from the inlet and outlet sections by a distance on the order of one-third of the fragment range in the gas, specific energy deposition steadily decreases with an increase in x for any y = const plane (Fig. 9.26а). At pressures higher than the optimum value, a reverse pattern emerges (Fig. 9.26c), with the exception of the near-wall regions, where the decrease in energy deposition with an increase in x is retained at all pressure values.

It follows from flow calculations in real NPL channels that the dependence of the main flow’s velocity upon the coordinate is close to linear (this fact is in full agreement with the results of the calculations performed in ref. [44]):

Un(x) « U0 + Bx. (9.149)

In this case, according to Eq. (9.146), we get:

Fig. 9.26 Distribution of specific energy deposition in a laser channel in the longitudinal direction at an argon pressure of (U0 = 4.5 m/s): P = 0.5 atm; (a) P = 0.9 atm; (b) P = 1.4 atm (c); (1) y = 0; (2) y = 0.4 cm; (3) y = 0.8 cm; (4) y = 0.9 cm

l.(_,) » + bSL

It follows from a comparison of the Eq. la(t) « Afat and relation (9.150) that the time required for a portion of the gas to traverse a distance of x from the channel inlet comes to

t

The latter is in agreement with the results obtained in ref. [43], in which

2E0ei^uq Т0Р0 cpd

where E0 is the kinetic energy of the uranium fission fragments, and e, is a certain effective value of the share of energy of the fission fragments that transmit to the gas. An analysis of the calculation results demonstrated that e, is slightly smaller than the ideal energy deposition efficiency, e0; however, the deviation of e, from e0 at all pressure and velocity values does not exceed 20 %. In the presence of sufficiently high velocities and weak fission intensities, when Bx/U0 << 1, we get a simple dependence

la(x) — A