A circuit with continuous transverse gas circulation was used in the LM-4 four — channel laser module operating together with BIGR reactor [41], within which a CW lasing mode was first achieved for NPLs [26] (see Sect. 6.1). The lasing lasted

1.5 s and ended with exciting neutron pulse termination.

The LM-4 apparatus consisted of four laser channels (Fig. 9.15) that were incorporated into a common gas loop and were separated from one another by plate-type radiators. The active length of the channels, determined by the dimension of the uranium layers along the optical axis, was LA = 1 m. The dimension of each channel in the direction perpendicular to the gas flow was d = 2 cm, while in the direction parallel to the flow, it was b = 6 cm (see Fig. 9.1). The average thickness of the uranium layer was <5u = 5 mg/cm2. This layer was covered with a protective aluminum film that had a thickness of SA = 0.5 mg/cm2. The LM-4 apparatus was irradiated by a neutron flux from a BIGR pulse reactor [42] with a duration of ~1.5 s. Thermal neutron flux density in the channel location at the pumping pulse peak reached 3.5 x 1014 cm~2 x s_1. The gas circulation system, based on motion of the hydropneumatic piston with a rectangular cross-section [41], ensured a gas velocity that was homogeneous along the length of the channel. It was switched on roughly ~0.3 s before the neutron pulse start and brought the gas velocity up to its maximum value of 4.5 m/s over this time interval. Gas circulation ceased immedi­ately upon neutron pulse termination. Under such conditions, after the lasing threshold is exceeded by roughly ~2-3 times, the shape of the laser pulse quite closely duplicates that of the pumping pulse.

Fig. 9.15 Transverse section of an LM-4 laser module together with a BIGR reactor: (1) aluminum substrate with a uranium layer; (2) radiator; (3) graphite; (4) casing; (5) gas pipeline

In order to investigate the effect of the gas circulation mode on the shaping of a laser pulse, two experiments were carried out in ref. [26] that involved an Ar-Xe mixture (70:1) at a pressure of 0.35 atm. During the first of these tests, the pumping pulse’s leading edge noticeably advanced the start time of gas circulation (delay of circulation start time); during the second test, the circulation system was activated long before the start of the neutron pulse and cut off the gas feed at the pumping pulse’s leading edge (premature switching of gas circulation). These experiments demonstrated that lasing does not occur until gas circulation begins, and con­versely, that lasing is terminated when circulation stops.

The physical model proposed in ref. [43], which describes the dependence of lasing active volume upon gas mixture composition and density, gas flow velocity, and specific power deposition, provides entirely satisfactory agreement with the experimental time characteristics of laser pulses. According to this model, over the period of time required for a portion of the gas to traverse a laser channel, if gas velocity at the inlet and fission intensity in the uranium layers are negligibly altered,

AUo(t) << Uo, Aqc(t) << qc, (9.90)

then the gas mass flow at each given moment in time can be roughly regarded as constant:

PoUo(to) ~ P (x, t) x U(x, t), (9.91)

where U(x, t), p (x, t) is the gas velocity and density values in the laser channel in a plane with a coordinate of x at a moment in time of t, averaged over the cross­section; Uo is the gas velocity at the channel inlet (which is homogeneous through­out the cross-section); to is the moment in time that the portion of gas under consideration enters the channel; and po is gas density at the channel inlet (which is kept constant). The average velocity and density values are determined by the equations

It is assumed that gas density, velocity, and temperature distribution will not be dependent upon the z coordinate, which is oriented along the system’s optical axis. The approach used reduces the description of the gas flow in the channel to a one-dimensional model.

During a gas flow in a plane-parallel channel, the length of which, b, is comparable to its transverse dimension, d, the pressure differential, ДP, between the channel inlet and outlet should be small (ДP/P << 1 (see the first section of this chapter)). Thus, from the condition of approximate pressure equality at any given point in the channel, P(x) ~ const, and the state equation for an ideal gas, we get

P0T0 = p(x, t)- T(x, t). (9.93)

Here, T0 is the temperature at the channel inlet, and T(x, t) is the temperature value in a plane with a coordinate of x at a moment in time of t, averaged over the channel cross-section, which, when small temperature deviations, AT(x, y, t) << T(x, t), are present within the confines of the transverse cross-section under consideration, is determined by a formula similar to Eq. (9.92).

It follows from correlations Eqs. (9.91) and (9.93) that

U(x, t)= ^ U0(t0). (9.94)

T 0

By virtue of the conditions in Eq. (9.90), the following heat balance correlation for a laser channel at each given moment in time should roughly hold true

x

where qS is the surface power density of the sources of energy deposition in the gas from both uranium layers. With allowance for Eq. (9.95), correlation Eq. (9.94) takes the form

U(x, t)

The energy deposition distribution of surface fission fragment sources is quite homogeneous, with the exception of the edge parts at the channel inlet and outlet. Ignoring the edge effects, we can assume that qS = const. Also, bearing in mind that U(x, t) = dx/dt, then from the latter equation, we get an equation that reflects, in a one-dimensional model, the relationship between the path traversed by a portion of the gas in the channel and the time over the course of which this path is traversed:

dx/dt = U0(t0) + Bx,

where B = (T0P0Cpd) 1q5.

At the initial condition of x(t0) = 0, the solution to this equation for the time it takes a portion of the gas to traverse a distance of x from the channel inlet, т = t — t0, within which the Eq. (9.92) yield the average density and velocity value distribu­tion, takes the form

In the special case of sufficiently large pumping velocities,

Bx/U0 < < 1, (9.97)

we get a simple dependence for the travel time

т = x/U0. (9.98)

When a laminar gas flow is present, its heat exchange with the laser cell walls occurs by means of conventional molecular heat conduction. If the o т time value is small enough that the condition l << d is satisfied for the transverse dimension of the near-wall region involved in heat removal, then from Eqs. (8.3) and (9.98) when the condition (9.97) is satisfied, we get

l(x) ~ v/ox/U0. (9.99)

If it is assumed that the development of the heat removal zone at the channel wall provided gas circulation occurs in accordance with the same regularities as in a cell that contains a stagnant gas, then in the general case, the laser active region volume per unit of laser channel length (along the z-axis) equals

b

0

where yA(x) = d/2 — l(x), while the relationship between the x coordinate and travel time, т, is determined by Eq. (9.96).

It is not difficult to show that in the special case when the l << d and inequality (9.97) are satisfied for the entire range of x, the laser active region volume is determined by the equation

where Ai is the proportionality factor in correlation (9.99). In the case of sealed (without gas circulation) laser cells, the formula l(t) = A^/at, which corresponds to correlation (9.99) written in the form l(x) = A^Jax/u0, as a rule satisfactorily describes the yA(t) dependence obtained from an accurate gasdynamic calculation over an interval of 3d/8 < y < d/2 (y varies over limits of 0 to d/2). However, for the specific alternative of a planar cell filled with Ar at P0 = 0.35 atm and T0 = 293 K, it allows the calculated dependence of yA(t) over an interval of d/4 < y < d/2 to be reproduced with an accuracy better than 6 %. Here, Ai« 3 for sealed cells (see Sect. 8.3), while the A factor can take on values one-and-a-half to two times smaller than for cells with gas circulation.

The model described was used in ref. [43] to calculate the dynamics of the development of a laser region’s active volume during experiments involving artificial desynchronization between the start of neutron pulse and gas-circulation system switching. This volume was found from Eq. (9.100) using the results of the calculation of active region size variation with the passage of time in an identical sealed cell. The relationship between the x coordinate in expression (9.100) and travel time, t, which corresponds to the duration of irradiation in a sealed cell, was determined using Eq. (9.96). The dependences of the neutron flux and gas flow velocity upon time needed for the calculations were assigned based on experimental data.

When the lasing threshold is exceeded, the active volume and the specific power deposition in the gas, which is proportional to the neutron flux, determine the laser output power at each given moment in time. The experimental dependences of neutron flux density, gas velocity, laser output power, as well as the calculated dependence of the product of active region volume times neutron flux density, upon time relative to their maximum values are presented in Fig. 9.16 for an alternative with delay of circulation start time. The results of the calculation of similar dependences for an alternative with premature switching of gas circulation are reflected in Fig. 9.17. The calculated cross-sections of the active volume (the clear area) and the zone involved in heat removal (the dark area) are shown in Fig. 9.18 for several successive moments in time that correspond to Figs. 9.16 and 9.17.

Fig. 9.16 Dependences of neutron flux density (1), gas velocity (2), laser output (3), and the product of gas active volume and neutron flux density (4) upon time for alternative with delay of circulation start time

a R. u. b R. u.

Fig. 9.17 Dependences of neutron flux density (1), gas velocity (2), lasing output (3), and the product of gas active volume and neutron flux density (4) upon time for alternative with premature switching of gas circulation

b

Fig. 9.18 Time dependences of the transverse cross-sections of the laser-channel active volume (clear area) and the zone involved in heat removal (dark area) for two alternatives: (a) delay of circulation start time, and (b) premature switching of gas circulation

Entirely satisfactory agreement is observed between the behavior of the exper­imental dependences of laser output (3) upon time and the curves (4) that reflect the value of the product of the neutron flux and the calculated active region volume values at each given moment in time. The difference in the slopes of the corresponding curves at the leading edge (Fig. 9.17) is explained by the fact that the lasing start has a threshold nature, unlike the process of the formation of a near­wall heat removal zone. Apparently, the medium small-signal gain is minimal at the moment of lasing start and begins to increase with an increase in power deposition. Therefore, the experimental curve of the dependence of laser output upon time runs
noticeably steeper. During these experiments, thermal neutron flux density at the lasing threshold was Ф « 9 x 1013 cm~2 x s_1, which corresponds to «0.26 r. u. (see Fig. 9.17).

Peculiarities of the Gas Flow in Flowing NPLs. Two-dimensional calculations of spatial inhomogeneities in NPL flowing channels were performed for the first time at the Idaho National Engineering Laboratory [44] using a specially developed gasdynamic program. The results of the calculations performed in ref. [44] were later repeated at the VNIIEF [45]. Because the VNIIEF did not have a gasdynamic program similar to the one used in ref. [44], a simpler calculation procedure was proposed that was based on a series of reasonably simplified physical models [46] (see also the subsection entitled “Calculations of Spatial Inhomogeneities in Flowing NPLs”). The difference in the density distribution calculation results obtained using this procedure and those of the similar calculations performed using the two-dimensional program [44], which makes strict allowance for gasdynamic and heat exchange processes in the entire gas volume, do not exceed ~2 % [45, 46].

One of the principal aspects of this procedure consists of the use of a physical model according to which the heat exchange processes in the bulk of the gas volume can be ignored. This model holds true, provided that

U0 > > ab/d2,

where a is the thermal diffusivity coefficient of the gas. Because the gas-flow rate in the NPLs under consideration comes to U0~10 m/s, then at typical channel transverse dimensions, b~6 cm and d~2 cm, as well as a rare gas thermal diffusivity of a ~0.2-0.6 cm2/s, the condition written above is virtually ensured.

In ref. [46], calculations were performed for a wide range of working gas pressure and velocity variations, which made it possible to optimize both the total energy deposition in the gas and the degree of specific energy deposition inhomo­geneity through the channel volume.

A weak facet of the calculation procedure used consists of its passive zone model, i. e., the near-wall region that does not take part in lasing. Numerical and theoretical investigations carried out during 1979-1984, which are described in detail in Chap. 8, revealed that the temperature profile has a clearly pronounced maximum and the density profile has a clearly pronounced minimum in sealed NPLs at times of t < 0.1 s. The latter is explained by the fact that in direct proximity to the laser channel side wall to which a uranium layer is deposited, a gas zone is formed from which the heat is intensely removed to the wall. This is called the passive (dead) zone.

It is also obvious that the same passive zone formation effect must take place in NPL flowing channels (where, as is generally known, specific energy release increases from the center to the side surfaces), the wall temperature of which is lower than the gas flow temperature. Accordingly, a proposition was advanced in ref. [43] that the development of a heat removal zone at the channel wall during gas flowing occurs qualitatively under the same laws as in a cell without gas flowing, i. e., the existence of a passive zone is proposed, the boundary of which is
determined by the density minimum and the temperature maximum. The relation­ship between temperature and density is unequivocally determined by a state equation. Because the pressure in the laser channel’s transverse cross-section is almost homogeneous (P « const), then in order to fully describe both the density and the temperature profile, it is sufficient to examine the dependence upon the coordinate of one of these parameters.

However, in an actual situation involving a gas flow, a boundary layer with a thickness of lv is formed near the wall, within which velocity increases from zero at the wall itself to a value equaling that in the flow core. Over the course of temperature equalization between the wall and the heated gas, hydrodynamic effects and heat conduction effects have a strong influence on one another [2].

For a compressible gas, temperature and velocity distribution near a planar wall in the presence of stationary motion is described by an equation system [2] that includes heat conduction, motion, and continuity equations

as well as a state equation

T = P/(y — 1)cvp.

Here, k is the heat conductivity coefficient; u is the longitudinal velocity compo­nent; v is the transverse velocity component; w is the specific power of the internal energy sources; and n is the dynamic viscosity coefficient.

The heat conductivity and dynamic viscosity coefficients of rare gases have a power dependence upon temperature [34]:

k = k0Tn; n = n0Tn. (9.106)

For these gases, n « 0.7 [34]. Even within the viscous and thermal boundary layers (as well as the passive zone, if it has not been eliminated), under conditions of actual powerful flowing NPLs, the absolute temperature values vary by roughly two times both in the longitudinal and the transverse directions. According to the relation (9.106), temperature differentials of this type will lead to variations of
~60 % in the heat conductivity and dynamic viscosity coefficients. Therefore, in order to find the dependence of viscous boundary layer thickness and the velocity distribution within its confines under actual NPL operating regimes, upon gas mixture flowing conditions and parameters, it is necessary to solve the system of essentially nonlinear differential equations in partial derivatives (9.102)-(9.106). Here, in addition to the boundary condition for velocity at the wall itself, it is correct to give the boundary condition far from the wall not at infinity (y = 1), as is frequently done in order to simplify certain problems, but rather in the plane of symmetry, i. e., at a finite distance of y = d/2. The same thing also pertains to temperature. At the wall itself, a Derichlet boundary condition, T(x, 0) = f1(x), or a Neumann boundary condition, dT(x, 0)/dy = f2(x), is given for temperature.

Thus, the solutions for the effective thickness of a viscous boundary layer, lv, and the velocity distribution therein, as well as for the width of the passive zone, la, and the temperature distribution therein, are functionals of the type

where T0 is the temperature value at the channel inlet (in these functionals, depending upon the conditions assigned at the wall, the temperature T(x,0) may be present instead of the derivative 3T(x,0)/3y. In addition, for simplicity’s sake, the dependence upon thermophysical parameters, especially upon the Prandtl number, is not reflected here). In point of fact, the problem is even more complex, because the distribution of specific sources in the gas will be dependent not only upon the fission intensity of the uranium nuclei in the uranium layers deposited to the channels walls, but also upon velocity and temperature distribution. A problem of this type for actual NPL modes can only be solved by means of numerical techniques involving the use of two-dimensional gasdynamic programs.

Of course, it may also be useful to a certain extent to obtain analytical solutions by introducing various simplifying assumptions, for example, of the T(x, y) — T0 << T0, v = const, and w = const types, etc. This helps mark the paths for optimizing energy depositions in the gas by means of gas pressure and velocity variations, an even to estimate share of passive zone volume (however small). But it must be said that any viscous boundary layer and passive zone models not based on the results of rigorous two-dimensional numerical calculations or direct experiments for NPLs operating at real power depositions cannot be regarded as correct.

From equation system (9.102)-(9.106), it is not difficult to obtain an integral equation for the passive zone. Actually, from continuity Eq. (9.104) and state Eq. (9.105), we get

dT

u + v
dx

Placing this result into Eq. (9.102) and integrating from zero to la with allowance for the fact that v(x,0) = 0 and that pT = p0T0 at P = const, we obtain

Possibility of Passive Zone Elimination

We will now examine the conditions under which a passive zone may be elimi­nated. Let us suppose that la = 0; it then follows from Eq. (9.108) that dT(x,0)/dy = 0, i. e., there is no heat flux at the wall. On the other hand, setting the heat flux at the wall to equal zero, then from Eq. (9.108), we get

du (x, y) dv (x, y) = w (x, y)

dx dy cpPfT 0

The latter equation means that when there is no heat flux at the wall, a passive zone can only exist if the specific heat source power distribution profile is similar to the sum of profiles of the derivative for x from the longitudinal velocity component and the derivative for y from the transverse component. It is obvious that the simulta­neous existence of such a coincidence over the entire extent of a flow in an NPL channel under normal operating conditions is unrealistic.

Thus, a necessary and sufficient condition for passive zone elimination consists of the heat flux at the gas and channel wall boundary equaling zero. Moreover, there will be no passive zone if the heat flux is directed from the wall to the gas (dT(x, 0)/dy > 0). This result makes it possible to estimate the channel wall heating needed for passive zone elimination. We will then perform an upper estimate. To this end, we will assume that the gas within a channel part from the inlet with a length of x over a time interval with a duration of t is in a state of rest. Here, its density is homogeneous and equals the gas density at the inlet, p0, while the specific energy deposition in the entire volume under consideration coincides with that in an infinitely expanded planar channel. In this model, the specific energy deposition near the channel wall can only be overstated, because in an actual instance when there is no passive zone, dp/dy < 0 at any y value. Consequently, specific energy deposition near the wall in an actual situation will also be lower than in the model under consideration. Furthermore, in a real case, gas density steadily decreases in the direction of x, dropping roughly half in size near outlet (see refs. [44, 46]), which also leads to a decrease in specific energy deposition.

During gas pumping through any given cross-section of x = const at any given instant in time, a new portion of gas flows in that is heated to a lesser extent than in the case when this gas remains motionless. The gas velocity in each cross-section increases from the wall to the center from zero to U(x). Therefore, the transverse temperature gradients when pumping is absent do not at least exceed the absolute value of the similar gradients when gas pumping is present. Consequently, the heat flux from the wall to the center in an unpumped model does not exceed the corresponding fluxes in a flowing system. Thus, in the nonflowing model under consideration, the wall temperature can only be overstated as compared to a flowing system.

A comparison of an actual flowing system and a model nonflowing system in a plane at a distance of x from the inlet can be made over a time interval of t = x/U (x), where U(x) is the average gas velocity over the transverse cross-section. It steadily increases with an increase in the distance from the inlet [44, 46]. However, the t time value can be determined in a simpler manner:

t = x/U0, (9.109)

knowingly overstating it in this instance, and accordingly, overstating the warm-up temperature.

So, ignoring longitudinal heat conduction, we get the following boundary value problem for the gas temperature distribution in a nonflowing model:

The boundary condition in Eq. (9.111) reflects the fact established above that a necessary and sufficient condition for the absence of a passive zone consists of the heat flux at the gas and solid wall boundary equaling zero.

Making the upper-bound estimate even more rigorous, we will now assume that the heat conductivity coefficient is equal to its minimum value everywhere, which corresponds to a value at the inlet of k(x, y, t) = k0, i. e., we ignore its dependence upon temperature in Eq. (9.106), thereby understating heat removal from the wall to the center of the channel. In this instance, Eq. (9.110) takes the form:

(9.116)

The upper temperature values of the channel wall at its contact surface with the gas that are needed for passive zone elimination were calculated using Eq. (9.116). The temperature values obtained for uranium fission intensity in the active layers, q = 2 x 1016 cm-3 x s-1, at an active layer thickness of Sv = 2.78 x 10—4 cm, as well as channel transverse dimensions of b = 6 cm and d = 2 cm, are presented in Fig. 9.19.

The specific power deposition in a gas was calculated using the results obtained in ref. [49]. The gas temperature at the channel inlet was given as equaling T0 = 293 K.

By way of illustrating the depth of heat exchange zone penetration into the gas, the dependences of the relative temperature increments, в = (T(x, y) — T0)/(T(x,0) — T0), and specific power deposition, m = w(x, y)W(x,0), upon the transverse coordinate, calculated at a point of x = b = 6 cm for helium at a velocity of u0 = 4.5 m/s, are presented in Fig. 9.20.

In order to achieve the stationary mode, it is necessary to ensure wall cooling through the use of a liquid heat-transfer agent, such as water, for heat exchange.

At P = 1 atm, the boiling point of water is TK = 100 °C. Water boiling in the external cooling channel can lead not only to the noticeable worsening of heat exchange, but also to vibrations and acoustic perturbations on the outer cooled surfaces of the laser channels that occur both during the formation and during the detachment of steam bubbles. The latter must inevitably lead to the transmission of the aforemen­tioned perturbations through the channel wall to the laser gas. Consequently, small- scale density inhomogeneities will occur, the result of which will be a deterioration of laser characteristics, or perhaps, simply lasing failure.

For a planar wall in the stationary mode, the temperature differential between its limiting surfaces equals

AT = QSw/kw, (9.117)

where Q is the density of the heat flux through the wall; Sw is the wall thickness; and kw is the wall heat conductivity coefficient. A uranium layer is deposited to the wall surface that adjoins the gas (see Fig. 9.1). The fission fragments carry a portion of the energy released in the uranium layer, which equals є, into the gas. The remaining portion of the energy (1 — e) in the form of heat is released within the layer itself. If the requirement of passive zone elimination, dT(x,0)/dy = 0, is satisfied—i. e., the heat flux from the wall (or more precisely, from the uranium layer) into the gas equals zero—the excess portion of the heat (1 — e) must then flow through the wall to the heat-transfer agent. In this instance, the heat flux density will be

Q = (1 — e)E0q8v, (9.118)

where E0 is the energy released in the single fission.

As in the preceding calculations, let us assume that q = 2 x 1016 cm—3 x s—1. When the argon velocity at the inlet is U0 = 4.5 m/s, then based on the previously cited upper-bound estimates, the wall temperature at the contact boundary with the gas will equal Tw(b) = 1,055 K. In order to avoid heavy water boiling near the outer wall
surface, the temperature differential at the wall must exceed ДT = Tw(b) — TK = 682 K. Pursuant to Eqs.(9.117) and (9.118)

КДТ

(1 — e)EQq8U at є«0.1 and for the thickness of a wall made from zirconium (kw = 0.21 W/cm x K), we get Sw = 1 cm. Similarly, for U0 = 9 m/s (Tw(b) = 696 K), we obtain 8w « 0.5 cm.

Cooling with an External Heat-Transfer Agent. In the stationary operating mode, if all the energy released in the laser channel is removed exclusively through the gas by means of its pumping, then at a thermal neutron flux density of Ф ~1014 — 1015 cm—2 s—1, even during transverse pumping, the requisite velocity, as follows from a balance equation of the (9.4) type, must be U0~100 m/s. However, as previously stated, heat removal through the outer surface of the laser channel substrate can be accomplished concurrently with gas pumping using the heat-transfer agent that washes around this substrate. But in the quasi-stationary mode, an appreciable portion of the heat released in the uranium layer can be absorbed in the substrate itself, as is the case in the LM-4 apparatus [41], or in the material mass directly adjacent to the substrate.

If the directions of movement of the external heat-transfer agent and the gas in the laser channel coincide, it is not difficult to establish the correlation between the geometric and thermophysical parameters of a system, during which the projection of the refractive index gradient in the y direction perpendicular to the uranium layer’s plane is (Vn)y < 0 in the entire gas volume, i. e., there is no near-wall passive zone. This can take place provided that (VT)y > 0 everywhere, including near the uranium layer surface. This means that the heat can only flow from the substrate to the gas. In order to implement this condition, it is essential that the heat fraction carried out of the channel by the gas is not less than є.

The total amount of energy carried away by the gas per unit of time through the cross-section of a laser channel with a coordinate of x per unit of channel length in the z direction equals

Q1 = dU0P0cP(r(x)— T0) • (9.119)

It is obvious that the realization of the condition (VT)y > 0 must satisfy the inequality

0

where qS is the power surface density of energy deposition in the gas from both of the uranium layers that is directly due to the escaping fission fragments,

qs = 2EqNv( оф Sus, (9.121)

where the angle brackets denote averaging over the neutron spectrum.

The total amount of energy per unit length in the z direction that the heat-transfer agent carries away per unit of time through the cross-section with a coordinate of x for each of the two external channels intended for the heat-transfer agent equals

Qi = d, UIoplocpl(T,(x, t) — To), (9.122)

Here, the l index indicates that the corresponding parameters are related to the external heat-transfer agent.

The amount of energy carried away per unit of time from a unit of uranium layer surface through the substrate to the external heat-transfer agent that is required in order to implement the condition (VT)y > 0 must not exceed

qi = HfEoNu(cfФ)Яи(1 — є), (9.123)

i. e.,

x

Here, ^f is the ratio of the thermal energy released directly into the uranium layer during the fission of a single uranium nucleus to the total kinetic energy of the fragments, E0.

Combining Eqs. (9.119)-(9.124) and assuming in this instance that the uranium layer and the fission density within the confines of the channel’s expanse along the direction of the x-axis are homogeneous, we obtain

Here, allowance is made for the fact that the gas density, p(x), can be perceptibly altered downstream; therefore, є can be dependent upon x.

When an oncoming or transverse gas and heat-transfer agent flow is present, the situation becomes considerably more complex. So, for example, if the heat flux at the gas and uranium layer contact boundary equals zero during an oncoming flow in a given transverse cross-section plane of the system under consideration, the heat flux density vector on different sides of the that plane will have an opposite direction.

Passive Zone Transverse Dimension. As previously indicated, it is not possible to obtain a precise solution for the width of a passive zone in a gas-flowing channel under actual pumping conditions. Nonetheless, even an approximate estimative description of its behavior would be of interest, which can be obtained using a series of simplifying propositions. As far as order of magnitude, the average travel time of a gas particle in a channel comes to т ~ b/U0. If S is flow line displacement in the transverse direction, the transverse velocity component is then v ~ S/т. We will assume that flow line displacement within the confines of a passive zone in the presence of the flows originating in NPLs with gas pumping is much smaller than this zone’s thickness (S << la), i. e., v << laU0/b. The transverse temperature gradient in the zone under consideration is estimated as THa therefore, as far as order of magnitude, the second addend in the braces on the left side of Eq. (9.102) comes to v(dT/dy) << U0T/b. For the first addend in these same braces, we get u(dT/dx)~U0T/b. Thus, the second addend in the braces can be ignored. In ref. [46] (see the ensuing subsection), the quasi-Lagrangian coordinates associated with the flow lines in the active zone, x and y0, are introduced. They are expressed through Eulerian coordinates by the correlation dy0U0p0 = U(x, y0)p(x, y0)dy(x, y0), where U is the gas velocity in a transverse cross-section with a coordinate of x for a flow line that has a Eulerian coordinate of y = y0 at the channel inlet. By analogy, introducing a simpler relationship between the Eulerian x, y coordinates and the new x, y’ coordinates,

dy(x y’) = PodУ’/p(x, y’),

and taking into account the neglect of the second addend on the left side of Eq. (9.102), we obtain

We will now examine the behavior of a gas near a planar wall, assuming for simplicity’s sake that the channel’s opposite wall stands an infinitely great distance away. The conditions on approaching the wall’s leading edge are:

T(0,y’) = T0; u(0,y’) = U0; p(0,y’) = p0. (9.128)

The boundary conditions in the gas itself at the passive zone boundary, where the temperature reaches a peak, are:

dT(x, la)/dy1 = 0. (9.129)

On the wall itself, we will assume that

T(x, 0) = 7o. (9.130)

Let us suppose that source power, W, per unit mass is constant:

W = w(x, y’)/p(x, y’) = const, (9.131)

while the gas velocity outside the confines of the viscous boundary layer is constant and equals Uq:

u(x, y’ > lv) = Uq. (9.132)

According to the model being described, because the heat exchange processes in the bulk of the gas volume—i. e., outside the confines of the passive zone—can be ignored, the temperature of each flow line before it intersects that passive zone will then equal

We will determine the thickness of the viscous boundary layer near the surface of a planar plate using the correlation [2]

lv = R^J vx/Uq, (9.134)

where R is a numerical coefficient, and v is the kinematic viscosity. According to one of the viscous boundary layer models most frequently used in practice, the velocity profile therein is presented in the form of a linear dependence [2]:

my — 0 s y s l-(x);

Uq, y > lv(x).

In this instance, pursuant to the impulse theorem, R = 3.46 [2].

We will now examine the successive intersection of the viscous boundary layer and the thermal passive zone under consideration by a given flow line (Fig. 9.21). We will avail ourselves of a model according to which the zone in question and the viscous layer have a clearly pronounced boundary. Let us assume that la is the coordinate of the boundary of the zone under consideration in a cross-section with a coordinate of xa. If lv > la, then a liquid jet with a transverse coordinate of y’ all the way up to the intersection with the viscous layer’s boundary will have a velocity of U0. Pursuant to Eq. (9.134), starting with a longitudinal coordinate of xv = y2Uq/R2v, the jet flows within a region where the velocity, u(x, y’), is deter­mined by the linear dependence of Eq. (9.135) upon y. For a similar flow line at

p = p0, Eq. (9.133) can be presented as (for simplicity’s sake, we will omit the prime marks associated with the y coordinate below):

We will employ the hypothesis of the similarity of viscous and thermal boundary layers, extending it to the passive zone as well. A comparison of calculations and experiments [43] confirm the latter in an entirely satisfactory manner. Let us suppose that

la (x) = Aolv(x). (9.137)

From expression (9.136) for y = la under the condition la < lv, we get

W A o

We will seek a solution to the problem using the following separation of variables for the transverse derivative from temperature:

dT (x y)/dy = f(xMx, y).

Pursuant to the boundary condition in Eq. (9.129), the ф function becomes zero at the passive zone boundary. At the wall itself, when y = 0, it can be assumed that ф = 1. We will present ф(х, y) as a function of a single combined variable, Y = y/la(x); i. e., ф^, y) = ф^). Then, pursuant to Eq. (9.139)

From Eqs. (9.138) and (9.140), we get

whence, according to Eq. (9.139),

Equation (9.127) is written in the new coordinates in the following manner:

p(x, Y)k(x, Y)d Y) = p20cpll(x)U(x, Y)d-TxYl — p20Wl2a(x).

Integrating it from zero to unity with allowance for the conditions (p(Y = 0) = 1 and q>(Y = 1) = 0, as well as the dependence u(x, Y) = A0U0Y, for the velocity within the confines of the passive zone, we get

where a = k(x,0)/cpp0. We introduce 1

Y2ip(Y)dY

0 then using Eqs. (9.134), (9.137), and (9.138) for A1 , we obtain an equation for the A0 coefficient:

— a0 +1= 0,

where Pr = v/a is the Prandtl number.

Analysis demonstrates that at any uniform dependence, q>(Y), that satisfies the conditions q>(Y = 0) = 1 and q>(Y = 1) = 0, Eq. (9.141) has two pairs of complex conjugate roots and two real positive roots, the first of which is smaller than unity, while the second is greater than unity. The first of the real roots, A0 < 1, has physical significance by virtue of the assumption made that la < ly. However, if we proceed under the assumption that la > ly, then operating under this same premise, we obtain A0 < 1 for Pr > 0.3 (Pr > 0.6 for rare gases), i. e., la < ly, which contradicts the basic assumption.

An analysis performed for the Prandtl numbers that correspond to rare gases revealed that the A0 coefficient may be found within limits of 0.5 < A0 < 0.67. Thus, according to the correlations used, Eqs. (9.134) and (9.137), the coefficient A = A0 x R x /¥r in a dependence of

la = Afaxjv.

falls within limits of 1.4 < A < 1.9. Therefore, a conclusion can be reached: unlike sealed lasers, the size of a passive zone in gas-flowing lasers is dependent upon the Prandtl number, i. e., it is determined not only by heat conduction, but also by viscosity, and is roughly one-and-a-half to two times smaller than its value for sealed lasers without gas circulation.