The shapes, dimensions, and structures of radiators may differ depending upon the operating mode, power density, and location of the laser system relative to the irradiation source. During operation in the pulse and quasi-pulse (ти~0.1-10 s) modes, a radiator may take the form of a simple porous structure that functions based on the principle of heat absorption by virtue of intrinsic heat capacity. It is obvious that in the stationary mode, an external heat-transfer agent should cool a radiator. The heat must be removed from this radiator by means of the heat-transfer agent washing over its end surfaces, which constitute a continuation of the outer surfaces of the uranium layer substrates. A radiator should consist of a set of plates

Fig. 9.1 Laser channel cross-section: (1) metal substrate; (2) uranium layer; (3) radiator

Heat-transfer agent

Fig. 9.2 Plate-type radiator directly adjacent to a laser channel and cooled by a liquid heat- transfer agent

(Fig. 9.2) or tubes with microchannels for flowing the cooling heat-transfer agent. The choice of potential engineering solutions is quite broad and will depend upon the conditions under which a specific laser system must operate.

It is not difficult to estimate the efficiency of the radiator under consideration using the results obtained from solving the problem of gas heat exchange with the channel walls. If the wall temperature of a channel as a portion of the gas passes through it can be considered constant, and if the flow itself is laminar, the depen­dence of the average gas temperature over the channel’s transverse cross-section upon the distance traversed, x, then takes the form [25]:

T(x)= TR + (To — Tr)J2 Ai exp — дЦ, (9.32)

where a is the thermal diffusivity coefficient of the gas; de is the equivalent diameter of the channel; U is the average gas velocity over the channel cross­section; T0 is the gas temperature at the channel inlet; and TR is the channel wall temperature. The A, and ft, values are presented in Table 9.2.

When the pressure of a helium-based gas mixture is P = 2 atm, a « 1 cm2/s. Let us assume that U = 10 m/s, while the distance between the plates is

S0 = Sr = 0.5 mm, which corresponds to an equivalent diameter of de = 1 mm (in the radiator channel, Re = U0de/v ~ 150, i. e., the flow is laminar). An estimate based on formula (9.32) demonstrates that in the presence of a plate length as low as x = bR = 3 cm, the difference between the initial gas temperature and the temper­ature of the radiator at its outlet drops 70 times.

The amount of heat that fission fragments transmit to the gas in a laser channel over a time frame of t comes to

G = q cLcdbLt,

where Lc is the length of the laser channel in the direction of the optical axis. The amount of heat absorbed by the radiator that is not cooled by the heat-transfer agent during heating from a temperature of TR to T0 is

G = mRLcdbRPRcR(T 0 — Tr)

(we assume that the height of the radiative cooler’s plates equals the height of the laser channel, d, in the direction of the y axis). Here, cR is the specific heat capacity of the radiator’s plate material; pR is this material’s density; and mR = Sr/(S0 + Sr ) is the radiator’s intermittency factor. It follows from these two equations that the radiator’s effective operating time is

RbRpRCR(TQ — Tr )

q cbL

For radiator made of aluminum plates at qc = 10 W/cm3, bR = 3 cm; bL = 6 cm, and (T0 — TR) = 100°K, operating time is t~6 s. These parameter values proved to be entirely acceptable for the radiators of the LM-4 laser module. The neutron excitation time during its operation in conjunction with a BIGR reactor was ~1.5 s [26].

A plate-type radiator is one alternative for the porous system used to intensify heat exchange processes [27—31]. The equations that describe temperature distri­bution in the components of a porous system in the stationary mode take the form [27—30]:

Gcp = hv(T — t), dx

d2T d2T

k + з/ — h*(T — T), (9-34)

where G is the specific mass flow of the component being flowed; cP is this component’s specific heat capacity; т is the gas temperature; hv is the volumetric inter-pore heat exchange intensity; T is the temperature of a solid porous compo­nent; and k is the heat conductivity coefficient of the solid component.

We will demonstrate that the temperature distribution in a plate-type radiator is described by equations similar to Eqs. (9.33) and (9.34).

Gas Cooling in a Radiator. We will select the x axis direction so that it coincides with the gas flow direction (see Figs. 9.1 and 9.2). Under actual conditions, the average gas temperature over the transverse cross-section in the direction of the z axis can change perceptibly at distances of not less than a few dozen radiator plate thickness values; therefore, the dependence of the parameters examined below upon the z coordinate is negligible. We will isolate a gas mass element, m, between the plates:

m = p(x, y, t)Al(x, y, t)Ad(x, y, t)80 = p0Al0Ad080 = const, (9.35)

where p(x, y, t) is the gas density at a point with coordinates of x and y at a moment in time of t; Al(x, y, t) is the length of the gas mass element; Ad(x, y, t) is its height; S0 is the distance between the radiator’s plates; Al0 is the length of the element at the radiator inlet; and Ad0 is its height at the inlet.

For most gases, including rare gases, the enthalpy of a unit mass of gas is determined as

w = cP jT,

where cP1 is the heat capacity at a constant pressure; therefore, at a point with coordinates of x and y for a moment in time of t, the enthalpy of the element under consideration is

w(x, y, t) = cpj Al(x, y, t)Ad(x, y, t)§0p(x, y, t)T(x, y, t).

Accordingly, the increase therein over a time frame of dt is

dT

dw = cPj A l(x, y, t) Ad (x, y, t) S0p (x, y, t) dt. (9.36)

The temperature of the gas mass element under consideration can change as a result of heat transmission to the preceding or subsequent elements, or as a result of heat exchange with the radiator’s plates. The effective distance over which heat propagates directly through a gas during a time interval of t is estimated by the correlation

l

where lg is the effective heat propagation distance and ag is the thermal diffusivity coefficient of the gas. The maximum time of gas residence in a radiator is

tm ^ L/u,

where L is the radiator dimension in the x direction and u is gas velocity. Over this time frame, the heat from the element under consideration propagates through the gas a distance of

lg (9.37)

At optimum pressures (P ~2 atm for helium and P ~0.5-0.9 atm for argon), ag < 1 cm2/s, L is on the order of a few centimeters [22], and, u > 10s_1. Placing these values into (9.37), we obtain lg < 10 2 cm, which is appreciably smaller than the longitudinal dimension of the gas in the radiator; i. e., longitudinal heat transfer in the gas should not play a noticeable role. However, the heat exchange of the mass element under consideration with the radiator’s plates is convective in nature, which, as is generally known, is considerably more efficient than conventional heat conduction.

The pressure drop, ДP, between the inlet and outlet of a channel formed by two parallel planes can be estimated using the formula [4]

Д P = 12rjUbR/8l, (9.38)

(n is the dynamic viscosity coefficient), according to which ДP ~ 10—3 atm for the helium cooling example considered above, which is much smaller than its absolute value of P = 2 atm. Therefore, the enthalpy decrease (—dw) should equal the amount of heat that the mass element under consideration transmits to two adjacent radiator plates. The amount of this heat is determined by the equation

dQ = Дl(x, y, t)Дd(x, y, t)q(x, y, t)dt, (9.39)

where q is the bilateral heat flux density at the gas contact boundary with the surfaces of the two radiator plates at a point with coordinates of x and y. During convective heat exchange,

q(x, y, t) = «{t(x, y, t) — Ts(x, y, t)g, (9.40)

where a is the heat transfer coefficient in a planar slit and Ts is the radiator’s plate surface temperature.

Using a minus sign (as the gas enthalpy decrease) to set Eq. (9.36) equal to expression (9.39) and substituting the result in Eq. (9.40), we obtain

Viewing the gas flow between the plates as the aggregate of the motion of jets with a height of Ad and a width of S0, then from the condition for mass flow maintenance, we write

u(x, y, t)p(x, y, t)Ad(x, y, t)S0 = MQP0 AdQSQ,

where u is the gas velocity component in the direction of the x axis and u0 is the gas velocity at the radiator inlet. With allowance for this correlation, Eq. (9.41) takes the form:

dx дт дт дт

= + u + V,

dt д t д x д y

where v is the gas velocity component in the direction of the y axis. An operating mode of practical interest is the one during which the following inequation is satisfied

дт дт

u » .

dx dt

Expression (9.44) means that the rate of change in the gas temperature as its location changes noticeably exceeds the rate of change in its temperature at any fixed point of a radiator with the passage of time. Furthermore, due to displacement restraint in the direction of the y axis by the radiator’s transverse dimension (see Fig. 9.2), gas jet shifting in this direction is small, so that

Taking this into account, Eq. (9.42) is written as дт

dx = —0{t(x, y, t) — Ts(x, y, t)g, (9.46)

where a0 is the thermal interaction coefficient of the gas with the radiator:

a

CP1P0 U0 80

Temperature Field in a Radiator Plate. The heat conduction equation for a plate takes the form:

where a is the plate’s thermal diffusivity coefficient, which can be regarded as almost constant for the temperature variation range in which we are interested.

We will introduce the average temperature over the transverse cross-section of a planar plate

where 8r is plate thickness. Integrating Eq. (9.48) for dz from zero to 8R/2, we obtain

From the condition of heat fluxes equality at the plate and gas contact boundary, we get

kR^T 0 = a1[T (x, y, t) — T(x, y, 8r /2, t)]. (9.51)

Here, kR is the plate’s heat conductivity coefficient and a1 is the gas’s heat transfer coefficient during heat exchange with a single planar surface. The latter is linked to the a coefficient by the correlation

ai = a/2. (9.52)

The plate’s maximum temperature differential in the transverse direction can be estimated as

dT(x, y, d/2, t) 8r dz 2

If this differential is much smaller than the difference in the gas temperature and the plate surface temperature

the T(x, y, d/2, t) temperature value on the right side of (9.51) can then be replaced with T (x, y, t). In this instance, placing Eqs. (9.51) into (9.50), we obtain

Here, taking Eq. (9.52) into account

It follows from a comparison of Eqs. (9.51) and (9.53) that Eq. (9.54) holds true given the condition that

SR < —. (9.56)

a

Accordingly, Eq. (9.46) can also be written in the form

^ ?) = — a0{T(x, y, t)— T(x, y, t)}. (9.57)

For the stationary case of dT/dt = 0, Eqs. (9.54) and (9.57) in their form fully agree with Eqs. (9.33) and (9.34), which describe heat exchange in a porous system (for simplicity’s sake, we will hereinafter omit the bar over the T, which denotes averaging Eq. (9.49)).

Boundary Conditions. Directly at the inlet end, as is also the case in a porous system [27], heat exchange between the radiator and the oncoming gas flow is virtually absent due to the latter’s negligible heat conduction; consequently,

The gas temperature at the inlet (x = 0) is a function of the y coordinate

t(0, y) = f (y).

During gas propagation along a radiator that is cooled by a liquid heat-transfer agent, its temperature can only decrease:

Ф > а У) < /max

where /max is the maximum gas temperature value at the radiator inlet. It is also obvious that the temperature of the radiator plates cannot exceed /max:

T(X > 0, y) < /max. (9.61)

At y = 0, symmetry condition must be satisfied

Convective heat exchange with the liquid heat-transfer agent takes place at the y = d/2 + Sw boundary:

T(d/2 + Sw) — Ti = q,/ah

where Sw is the thickness of the radiator’s substrate; Tl is the temperature of the liquid heat-transfer agent; ql is the heat flux density at the y = d/2 + Sw boundary; and al is the heat transfer coefficient at the y = d/2 + Sw boundary.

The temperature differential throughout the thickness of the radiator’s substrate (the radiator’s substrate and its plates may be made from different materials) equals

T(d/2) — T(d/2 + Sw) = qSw, (9.64)

kw

where kw is the heat conductivity coefficient of the radiator’s substrate. The heat flux at the radiator’s plate and substrate contact boundary is

k dT(d/2)

qi = — kR dy

The boundary condition at the radiator’s plate contact boundary in the presence of y = d/2 follows from Eqs. (9.63)-(9.65)

where

Approximation o/ an Ideal Radiator. The simplest and apparently the most conve­nient for elementary estimates and purely qualitative arguments is an “ideal”
radiator approximation: gas cooling is examined in a situation when the plate temperature variations of the radiator itself are sufficiently small that their effect can be ignored in Eq. (9.57); i. e., the temperature, T, can be regarded as constant and homogeneous. In the stationary case, when the heat transfer coefficient, a, is not dependent upon time, from Eq. (9.57), we get

X

ao(x)dx

0

The gas-flow velocity in the laser channel is U ~ 10 m/s and the width of the gap between the radiator’s plates, like the thickness of the plates themselves, is S0~ SR ~1 mm. Consequently, the gas velocity in the radiator is u ~20 m/s. The Reynolds number for a flow of this type is

ude

Re = — — 700,

V

where de = 2£0 is the equivalent diameter of the gap between the plates and v is the kinematic viscosity of the gas (v ~0.6 cm2/s). This is appreciably lower than the critical value of Rec ~ 2,300, above which the flow is not laminar, but turbulent. The heat transfer coefficient a is determined by way of the Nusselt number:

Nu

a = kg— , (9.69)

de

where kg is the heat conductivity coefficient of the gas. In the presence of a laminar flow, the Nu value in the initial part of the flow decreases with distance from the inlet into the slit between the plates to a constant value in the stabilized section [25, 32]. For a planar slit formed by two parallel planes, the distance within which the stabilized heat transfer part begins is determined by the inequality [25]

x > ud2e/70ag. (9.70)

Besides, if the channel wall temperature is constant, then Nu = 7.5 within the stabilized part. A power law dependence is recommended for the initial part [25]

Nu = 1.85(ud2/agx)1/3. (9.71)

A calculation of the & = (t—t) parameter (Curve 1) using the formula (9.86) with the recommended Nu(x) dependence [25] is presented in Fig. 9.3. There, too, a similar calculation (Curve 2) using formula (9.32) is offered. The calculations were performed for helium at P = 2 atm and t = 400 K; the gas velocity was u0 = 10 m/s,

Fig. 9.3 Dependence of the reduced gas temperature upon the distance traversed within a radiator: (1) a calculation using formula (9.68) in the presence of a variable Nu value; (2) a calculation using formula (9.32); (3) a calculation using formula (9.68) at Nu = 7.5

while the distance between the plates and their thickness came to S0 = Sr = 0.5 mm, which corresponds to an equivalent diameter of de = 1 mm. For comparison, a similar dependence (Curve 3), obtained using formula (9.68) under the assumption that the Nu number is constant and equals 7.5 over the entire expanse of the radiator, is shown in Fig. 9.3. According to Eq. (9.70), the expanse of the stabilization part is ~2 mm. It is obvious that gas cooling efficiency can be calculated with quite good accuracy using a constant Nusselt number to determine the heat transfer coefficient a.

Real Radiator. We will now examine the stationary problem дт/dt = dT/dt = 0. It is not difficult to reduce equation system (9.54) and (9.57), with boundary conditions (9.58)-(9.62) and (9.66), to a single linear equation in third-order partial derivatives

d3r d3 r d2r d2r dr

dx3 + dxd + ao dx2 + ao dy2 — Hdx = 0

and boundary conditions

where

r(x, y) = t(x, y) — T,. (9.78)

The solution to problem (9.72)-(9.78) takes the form:

1 Cn /2в

r(x, y) = n {s2„exp(s1„x) — S1„exp(s2„x)} cos ny. (9.79)

n=1 S2n — S1n d

Here, the range of problem eigenvalues is determined by the transcendental equation

hd

tg P — = ; n

2Pn

while the s1n and s1n are the negative roots of the characteristic equation

s3 + a0s2 — (Xn + H)s — a0Xn = 0, (9.80)

where

4en

d2 (Generally speaking, Eq. (9.80) has three different real roots. For the actual thermophysical parameter values of the radiator system under consideration, two of them are always negative, while one is positive and increases without limit as the n parameter value increases).

The Cn coefficients take the form:

Here, ||Фп ||2 = d + 8^T sin2вп is the square of the norm.

Thus, according to Eq. (9.78), gas temperature distribution takes the form

1′ cn (2в

t(x, y) = T, + — {s2nexp(s1nx) — S1nexp(s2nx)} cos — ny, (9.81)

n=1 S2n — S1n d

consequently, the temperature distribution in the radiator’s plates is determined by the equation

T(x, y) = t(x, y) + — C”S1”S2” {exp(s1„x) — exp(s2„x)} cos 2впу. (9.82)

«0 n=1 s2n s1n d

The series in Eqs. (9.81) and (9.82) quickly converge. So, the results of a calcula­tion using ten terms of sum in these equations differ by at least 0.4 % from the results obtained with allowance for 20 terms.

By way of illustration, the first of the ten eigenvalues of fin = d^[X~n/2 and the sin roots of Eq. (9.80) that correspond to them are presented in Table 9.3. The values cited correspond to helium at p0 = 4.6 x 10-4 g/cm3 and P = 2.8 atm, pumped at a rate of u0 = 9 m/s through an aluminum radiator with plate dimensions of d x SR = 2 x 0.05 cm. The thickness of the gap between the plates is S0 = 0.05 cm; the radiator’s substrate (also aluminum) has a thickness of Sw = 0.16 cm. The heat transfer coefficient of the liquid heat-transfer agent is at = 1.7 W/cm2 x K.

The calculations results for the aluminum plates of a radiator with a thickness of SR = 0.5 mm are presented in Figs. 9.4, 9.5, 9.6, 9.7, 9.8, 9.9, and 9.10. The width of the slits between the plates is S0 = 0.5 mm. It was assumed that the laser channel within which gas mixture excitation by fission fragments occurs has transverse dimensions of d x b = 2 x 6 cm, that the thickness of the uranium layers is 5U = 2.78 x 10-4 cm, that the intensity of uranium nuclear fission is qU = 2 x 1016 cm-2 x s-1, and that the gas temperature at the laser channel inlet is Tg0 = 293 K. During the calculations, it was assumed that the heat-transfer agent’s heat transfer coefficient is a = 1.6 W/cm2 K, which corresponds to water at T; = 293 K, pumped at a rate of 5 m/s through an external cooling channel (see Fig. 9.2), the transverse dimension of which comes to 3 cm. The thickness of the radiator’s aluminum substrate was taken to equal = 1.6 mm.

Figures 9.4 and 9.5 reflect the calculation results for the argon temperature distribution within a radiator in the transverse and longitudinal directions following gas transition from a laser channel within which the gas pressure amounted to

0. 9 atm, while gas-flow velocity at this channel’s inlet was U0 = 4.5 m/s. At the outlet from the laser channel, gas-flow velocity as a result of heating by the fission fragments equals 10.78 m/s, which, in the presence of the SR and S0 dimensions indicated above, corresponds to an initial gas velocity in the radiator of u0 = 21.56 m/s. The gas temperature distribution at the radiator inlet (the channel outlet), t(0, y) = f(y), corresponds to Curve 1 in Fig. 9.4.

The heat transfer coefficient, a, determined by Eq. (9.69), will be dependent upon the gas heat conductivity coefficient, kg, which is a function of temperature:

For helium, k0 = 0.159 W/m x K, Tk = 320 K, and n « 0.69, while for argon, k0 = 0.0177 W/m x K, Tk = 300 K, and n« 0.71 [33—35]. Therefore, the heat transfer coefficient is reduced when the gas is cooled as it passes through the radiative cooler, leading to a slight decrease in the thermal interaction coefficient,

n	Pn	s1n	s2n	s3n
1	0.7677	-2.137	-0.336	1.115
2	3.3589	-3.685	-1.184	3.511
3	6.3986	-6.540	-1.310	6.492
4	9.5026	-9.591	-1.336	9.569
5	12.6250	-12.689	-1.346	12.677
6	15.7550	-15.805	-1.350	15.797
7	18.8888	-18.930	-1.353	18.925
8	22.0248	-22.060	-1.354	22.056
9	25.1622	-25.193	-1.355	25.190
10	28.3005	-28.327	-1.356	28.325

Table 9.3 Eigenvalues of and the roots of Eq. (9.80) that correspond to them

Fig. 9.5 Argon temperature distribution in the longitudinal direction at u0 = 21.56 m/s (U0 = 4.5 m/s): (1) у = 1 cm; (2) у = 0.5 cm;

(3) у = 0; solid lines are a real radiator; broken lines are an ideal radiator

a0, and the H parameter. During the performance of the calculations described above, a heat conductivity coefficient value was used that corresponded to the gas temperature at the radiator inlet, averaged over the transverse cross-section.

Comparative calculations were performed in order to check the extent of the influence of the thermal conductivity coefficient decrease, during which the heat conductivity coefficient was taken for the average gas temperature value (over the transverse cross-section) in a cross-section of x = 3 cm. The largest deviations occur in the case of argon; however, even they do not exceed в ~ 30 % when the relative deviations are calculated using the formula

1,100 1,000 900 800 700 600 500 400 300

=

T1 — Ti

where t1 is the temperature calculation at kg, determined by means of the average temperature at the inlet, and t2 is the temperature calculation at kg, determined by means of the average temperature in a cross-section of x = 3 cm.

The differences between the gas temperature and the radiator plate temperature for helium and argon are shown in Figs. 9.6 and 9.7. As is apparent from these graphs, the differences reach hundreds of degrees at distance of about 1-2 cm from the inlet, which testifies the invalidity of using the local thermal equilibrium model,

t, K

t «T, for the radiators under consideration, that is suggested in certain cases involving the calculation of heat exchange in porous systems [27].

The intensity of gas cooling will be substantively dependent upon the thickness of the slit gap between the radiator’s plates. The calculation results for longitudinal argon temperature distribution at S0 = 0.2 and 0.5 mm (Sr = 0.8 mm and 0.5 mm, respectively) are shown in Fig. 9.8 for comparison.

Presented in Fig. 9.9 is the dependence of the argon temperature at a distance of x = 3 cm from the radiator’s inlet upon slit width, S0, for the plate lattice spacing, Л, determined as

Л = <50 + Sr, (9.83)

Figure 9.9 demonstrates that it is possible to select a radiator length and slit width such that the temperature at the outlet will be homogeneous.

The calculated dependence of argon temperature upon the radiator spacing, Л, at a distance of x = 3 cm (for y = 0 cm) from the radiator inlet is shown in Fig. 9.10. The dependence presented suggests the possibility of performing radiator optimi­zation on the lattice spacing and the width of the gap between the plates.

Yet another possibility for the enhancement of radiator efficiency consists of expanding its transverse dimension along the y-axis immediately behind the inlet into the radiator itself. Indeed, the gas mass flow in the stationary mode equals G = p0u050d, so it follows from Eqs. (9.47) and (9.69) that

kgNu d 2cpG S0

Here, we took into account that de = 250 for a slit. By leaving the mass flow, G, and the slit width, 50, unchanged, but increasing the d dimension after the inlet, we achieve an increase in the thermal interaction coefficient, a0.

Sequential Circuit of Laser Channels and Radiators. In prospective multichannel laser systems that consist of a gas path made up of a sequential circuit of alternating laser channels and radiators (on the order of 25-50 channels and radiators), the longitudinal dimension of the radiators must be limited. The calcu­lations presented in the preceding subsection revealed that when radiators with a length of L = 3 cm (identical to those employed in the LM-4 quasi-pulse laser setup) are used in stationary multichannel laser systems, the gas temperature increase attained in the laser channel, t(0, y) — 7), is reduced by a total of two to three times. In this regard, a question arises: What temperature will the gas have after traversing a sequential circuit composed of N laser channels and N radiators?

As the calculation results demonstrated (see above), at distances of x > 3 cm from the inlet to the radiators, gas temperature distribution in the transverse direction is close to homogeneous; thus, for the sake of simplifying the description, we will assume that the aforementioned gas temperature distribution at the outlet for radiators with a length of L > 3 cm is scarcely dependent at all on the y coordi­nate. It follows from the solution to Eqn. (9.81) and the formula for Cn coefficients that the gas temperature at the first radiator’s outlet will be

tri(L) « Tl + <p(L) — и(Ь)Гі. (9.85)

Here, the following designations are introduced:

d/2

f (£) cos 2dn £ d£

1 d 2 в

Ф (L)~ 0 2 cos — — y {s2nexp(sinL) — sinexp(s2nL)};

n=1 II Фп Ir(s2n — sin) d

H (V)K, 2 2 n cos ddiy {s2nexp(sinL) s1nexp(s2nL)}

2-=i fink Фn Ir(s2n — sin) d

Let us suppose that the gas temperature at the first laser channel’s inlet equals t0, while at its outlet (at the first radiator’s inlet), ti(0, y) = f(y); i. e., the temperature increment after the laser channel has been traversed comes to

My) = f (y) — t0.

If we assume that all the channels in a common laser-radiator circuit are identical and that they operate under identical conditions, fission fragment energy deposition in the gas within each channel will then be identical, and accordingly, the temper­ature increments, Дт(y), will also be roughly identical.

Behind the first radiator, the gas temperature increment relative to the first channel’s inlet temperature, t0, is

Atri = T1 — T0 = Ti + ф — yTi — T0.

Because the gas temperature at the second channel’s inlet equals t0 + Atr1, while the temperature increment in each channel is identical, the temperature distribution at the second radiator’s inlet takes the form

f2(y) = f (y) + Атяь

then in accordance with the solution to Eqs. (9.81) and (9.85), at the second radiator’ s outlet, we get

тr2 = Ti + Ф — yTi + y Atr = (1 — y2) Ti + (1 — y) ф — уто,

i. e., the temperature increment at the second radiator’s outlet relative to the initial temperature, t0, is

AtR2 = (1 — y2)Ti + (1 + у)ф — (1 + y)T0.

For the outlet from the Nth radiator, we obtain

N

AtRn = l1 — yN)Ti + (Ф — T0) ym.

m=0

The sum on the right side of Eq. (9.86) is the sum of a decreasing geometric progression with a geometric ratio of y. Therefore, taking Eq. (9.85) into account, we get

The limiting value of AtRn (at N! 1) is

Atri = 1—yATR1. (9.88)

In the case of argon for a radiator with a length of L = 3 cm at a plate thickness of S0 = 0.5 mm and a gap thickness between plates of SR = 0.5 mm, ц = 0.358, while for helium, ц = 0.348. The limiting values of the e1 = AtRi/Atr1 ratios for argon and helium equal 1.558 and 1.534, respectively. The eN = Atrn/Atr1 ratios for these gases at N = 4 come to e4Ar = 1.532 and e4He = 1.511, which differ from the limiting value by a total of 1.5 %. Thus, starting with the fifth channel, all the lasers and radiators will operate under virtually identical conditions in the stationary mode.

It is useful to note that at S0 = 1 mm and SR = 1 mm (L = 3 cm), a similar calculation yields elAr = 3.497 and elHe = 2.513. Here, for N = 8, e8Ar = 3.26 and e8He = 2.469, which differ from the limiting value for argon by 6.8 % and for helium by 1.7 %. Technologically, it is quite a bit simpler to fabricate radiators with a plate thickness of S0 = 1 mm and a gap between plates of SR = 1 mm; therefore, in a multichannel apparatus with a large number (~50) laser channels sequentially connected in the gas path, it would perhaps be reasonable to give preference to radiators of this type.

Flow Behind a Radiator. The gas flow behind a radiator’s outlet was studied on the LUNA-2P setup [23, 24], the gas circuit of which included two laser cells with planar uranium layers (see Fig. 2.13). Three identical heat exchangers (radiators) were located on both sides of the cells: the first was at the first cell’s inlet; the second was between the cells; and the third was at the second cell’s outlet. The gas circulation system ensured the successive passage of the gas through the radiators and the laser cells. Laser cell active volume was limited by two planar aluminum plates with dimensions of 100 x 6 cm, which were positioned parallel to one another at a distance of d = 2 cm. Thin 235U oxide layers with a thickness 2.8 mg/cm2 were deposited to the inner surfaces of the plates. The radiators were made from rectangular aluminum plates with a thickness of S = 0.5 mm and a length along the gas flow of 3 cm. The distance between the plates came to 0.5 mm, while the height of the gas flow in the radiators was 2 cm. A laser channel scheme is presented in Fig. 9.1.

The LUNA-2P setup was irradiated by a pulsed thermal neutral flux from a VIR-2 M reactor that had a half-height pulse duration of ~3 ms. The thermal neutron flux density at the pulse peak, averaged over the laser cell’s active length (1 m), was 2.2 x 1015 cm~2 x s_1. The conditions and design of the experiments were the same as in ref. [36], where optical inhomogeneities in NPLs without gas circulation were studied (see Sects. 7.4 and 8.1). Specific energy deposition at the optimum gas pressures (PHe = 2 atm and PAr = 0.5 atm) came to 0.067 J/cm3.

Optical inhomogeneity measurements in a laser channel with gas circulation were performed using a Mach-Zehnder interferometer at a He-Ne laser wavelength of X = 0.63 pm under conditions of a reactor radiation background and slight mechanical fluctuations in the interferometer’s mirrors. In order to increase the measurement accuracy to 0.1 X, a comparative channel was used in the interferom­eter’s optical circuit. The fringe pattern was recorded using a standard SFR-2 M camera with a pulse feed system developed by the authors of refs. [23, 24], which expanded the recording time range to 0.1 s with a resolution no worse than 50 ps. Investigations were performed with helium, argon and air in the presence of a gas flow rate of up to 12 m/s in the laser channel, both at atmospheric pressure and at pressures selected in such a manner that the relative width of the gas gap between the uranium layers, D = d/R (where R is the range of an average uranium fission fragment in a gas of a given type) was close to the optimum value, D « 0.45 [37] (see Sect. 7.5).

A typical interferogram of the optical inhomogeneities in a laser channel at the time of neutron pulse termination, as well as the results of its processing, are

presented in Figs. 9.11 and 9.12. The interferogram was divided into three sections: the operating channel is in the center; and the comparative channels are above and below. It was experimentally shown in ref. [36] (see also Sects. 7.4 and 8.1) that when such an excitation mode is used, gas density redistribution in a laser is of the quasi-equilibrium type. Here, the acoustic waves that are usually observed in pulsed lasers are not present in the medium.

In sealed lasers (without gas circulation) during the heating of the active medium, its expansion into the buffer volume occurs, in addition to which a density profile is formed that corresponds to an energy deposition inhomogeneity. The gas circulation mode contributes its own specific features to the nature of the optical

inhomogeneities that originate. In the cross-section perpendicular to the uranium plate plane (Fig. 9.11), as in the case of a gas at rest, a positive lens originates that has a close-to-parabolic profile. When gas circulation is used, the parabola’s steepness increases downstream.

The distribution of refractive index deviations from the initial value, n0, in the cross-section parallel to the plate plane (along the x-axis in the plane of y = 0) is shown in Fig. 9.12. During exposure to an exciting pulse, the gas in the channel at a given velocity is shifted a total of 3 cm from the inlet radiator. Within this part (0 < х < 3 cm), the variation of the refractive index along the х-axis is almost linear in nature. The remaining portion of the gas in the region of х > 3 cm is present in the channel over the entire course of irradiation; therefore, as it approaches the outlet radiator, the dependence of the refractive index upon х becomes increasingly flat and reaches a constant value roughly ~2 cm before the outlet.

We note that the interferogram in Fig. 9.11 reflects a transient process in gas density redistribution, because the duration of the exciting pulse in these experi­ments was comparable to the time required for a portion of the gas to pass through the excitation zone.

In refs. [38, 39], calculations demonstrated that gas flow perturbation must occur at the outlet of a plate-type radiator, during which a pair of adjacent symmetrical stable vortex structures will be formed for each rectangular plate. The lateral dimen­sions of these vortices will decrease downstream and their dimension relative to plate thickness will be proportional to the Reynolds number. If the distance between the plates equals their thickness, the dimension will then be determined by the correlation

xw « 0.09 x Цк. (9,89)

The circulating motion of the gas in the vortices will lead to convective mixing, which will necessarily bring about the intensification of heat exchange processes as compared to conventional molecular heat conduction.

The laser sounding of an active medium in the gas flowing mode revealed that the fringe pattern near the radiator outlet disappeared due to sounding beam refraction on the radiator’s plates. The size of this region (~1 cm) and the large refractive index gradient value (~5 x 10~5 cm-1) confirm the proposition concerning increased thermal diffusivity near a radiator’s outlet. The dimension of the region described, xw, should increase with an increase in the plates’ trans­verse dimension and in gas flow velocity, and will be dependent upon the gas type.

In order to determine the thermal diffusivity associated with the origination of vortex structures at a radiator outlet, investigations were performed during which a nickel-chromium filament with a diameter of 0.2 mm and a length of 25 cm was placed in a laser channel at a distance of 3 mm from a radiator outlet. The filament was heated by an electric current, thereby creating a local thermal perturbation that extended downstream in the steady-state gas flow. The experimental procedure was based on the measurement of the temperature fields created in the cell’s volume using an interferometer. Interferograms were recorded using a He-Ne (LG-38) laser

Fig. 9.13 Interference fringe displacement as a function of height during filament heating: (1) x = 1 cm; (2) x = 3 cm; (3) x = 4 cm; (4) x = 5 cm

radiation and a Michelson interferometer. An SFR-2 M high-speed camera regis­tered the fringe patterns. The nature of heat exchange between the gas flow and the uranium layer substrate was studied separately. To this end, one substrate with a length of 25 cm was heated by AT = 40 °C using hot water that flowed inside it. The experiments described were carried out without neutron irradiation, i. e., there was no energy release in the uranium layers themselves.

A typical interferogram of the gas density perturbations that originate during filament heating in a helium flow (U = 8 m/s, Р = 2.2 atm, and S = 0.5 mm), together with the results of its processing, are shown in Fig. 9.13. The thermal perturbation had a Gaussian shape and expanded downstream, during which its peak value was reduced. Similar data that correspond to the same gas flow param­eters for an experiment involving substrate heating are presented in Fig. 9.14.

The experimental results made it possible to estimate the variation of effective thermal diffusivity coefficient along the gas flow. At the radiator outlet, it exceeds the molecular heat diffusivity coefficient by roughly one order and is reduced twofold at a distance of ~xw from the radiator outlet. This constitutes confirmation of the existence of vortex structures behind the radiator outlet, the transverse dimension of which is decreased downstream.

Fig. 9.14 Interference fringe displacement as a function of length during substrate heating: (1) y = —9.5 mm; (2) y = — 8 mm; (3) y = —6.5 mm

Based on the experimental results, an attempt was made to verify the hypothesis that the flow behind a radiator is turbulent in nature. The numerical analysis of a flow of this type did not present particular difficulties, because the temperature perturbation distribution behind the heated filament obtained during the experi­ments had a Gaussian shape. At first glance, this circumstance corresponds to the behavior of the heat wake from a thin linear source in a stationary turbulent flow created by artificially introducing a turbulizing wire grating at a certain distance ahead of the filament [40]. In this instance, thermal perturbation washout with distance from the source occurs in such a way that the maximum value and the half­height width of the perturbation peak are unequivocally linked to the turbulent heat diffusivity coefficient. The numerical analysis results demonstrated that when such allowance is made, the turbulent heat diffusivity coefficients, measured in cm2/s, are decreased downstream from a few tenths to order-of-unity values at the channel outlet. They will be dependent upon gas velocity and the transverse dimensions of the radiator plates. However, the coefficients obtained from an analysis of the downstream rate of variation in the maximum local thermal perturbation value proved to be several times higher than the similar coefficients found from this perturbation’s peak half-width washout rates. This fact suggests that it is impossible to regard the flow behind a radiator as a steady-state turbulent flow, and that the thermal diffusivity coefficient should be determined experimentally.

Experiments with a heated substrate revealed that the gas region near the uranium layers involved in heat exchange with the wall expands downstream (Fig. 9.14). It transverse dimension is roughly proportional to ~ ^/x (x is the distance from the cell inlet) and is dependent upon gas velocity, density and type. Depending upon the conditions in place, the full volume of the heat exchange zone in the cell may involve 10-30 % of the gas’s active volume. In ignoring the viscous boundary layer’s influence, it is not difficult to explain this dependence. Here, it can be roughly assumed that the gas velocity is homogeneous throughout the channel cross-section and equals U. We will isolate a plane with a coordinate of x in the transverse cross-section of the gas. Let us suppose that each such plane up to a moment in time of t = 0 is situated outside the channel’s confines. The time that this plane is located within the channel itself, including the region involved in heat conduction, equals to x/U. Over this time frame, the thickness of the region involved in near-wall heat exchange in the plane under consideration pursuant to Eq. (8.3) reaches a dimension of

where aeff is effective thermal diffusivity near the channel wall.

Thus, the significant role of vortex structures at the radiator outlet during heat conduction processes within a laser cell was demonstrated during the experiments. The most negative aspect consists of the considerable gas layer thickness near the substrate involved in the heat transfer process, as a result of which the lasing region
can only occupy a portion of the cell’s cross-section. In order to diminish this effect, it is necessary to reduce the size and intensity of the vortex structures in the gas at the radiator outlet by tapering its edges and bringing the temperature of the gas exiting the radiator as close as possible to that of the uranium layer.