Chapter 2

Space Nuclear Power Systems 1989 Edited by M. S. EI-Genk and M. D. Hoover © Orbit Book Company, Malabar, FL, 1992

Chapter 2 Ultrahigh Heat Flux Systems Using Curved Surface Subcooled Nucleate Boiling: Heat Transfer and Fluid Flow Studies for a Planar Collector Arthur H. Iversen

Stephen Whitaker

Coriolis Corporation Campbell, CA 95008

Department of Chemical Engineering University of California Davis, CA 95616

Curved surfaces can sometimes be incorporated directly into heat transfer systems; however, their use with large planar heat sinks may require a moderately complicated fluid distribution system. Our theoretical and experimental fluid flow studies describe one system that successfully incorporates curved surface nucleate boiling into a planar heat transfer surface. We describe the use of curved surface s~bcooled nucleate boiling to generate exceptionally high heat fluxes under flow conditions that require relatively small pressure drops. Our previous experimental studies have provided a critical heat flux of 6.3 kW/ cm 2 using the fluorocarbon FC 77 as a coolant. The coolant pressure drop for this condition was only 18.6 x IO J Pa per centimeter, thus indicating that higher heat fluxes can be achieved at moderate pressure drops. To illustrate the efficiency of curved flows relative to linear flows, we note that the estimated pressure drop required to achieve a critical heat flux of 6.3 kW/cm2 using linear flow is 27.3 X 106 Pa per centimeter. While there is uncertainty in this estimate, the disparity between 18.6 x IOJ Pa and 27.3 x 106 Pa clearly indicates the advantage of using curved surface boiling. This is especially true for space operations where minimizing the pumping power requirements is an important consideration.

Introduction

face. This increase in the body force can be generated by flow past a concave surface.

It is a well established fact that extremely high rates of heat transfer are obtained during subcooled nucleate boiling. It is also well known that these high rates originate with the intense liquid motion that is generated by the growth and subsequent departure of the vapor bubbles from the heat transfer surface. These vapor bubbles, which generate the desired fluid motion, are also the source of the boiling crisis known as the critical heat flux. As the heat flux rises, the number of active nucleating sites increases, the concentration of vapor bubbles in the neighborhood of the heat transfer surface increases, and bubble coalescence leads to the formation of a stable vapor film. Under these circumstances, a system operating at a fixed heat flux will usually fail owing to system burnout. If the rate at which vapor bubbles are removed from the heat transfer surface is increased, the onset of film boiling can be delayed and higher heat fluxes can be achieved. There are two principal mechanisms which can be used to enhance the rate of removal of vapor bubbles:

Shear

Critical heat flux phenomena have recently been reviewed by Katto (1985) for a wide variety of flow configurations. While the critical heat flux depends on the type of flow (internal, external, or jet flow, for example) and the nature of the heated surface (flat, curved, finite, unbounded, and so on), one can conclude that the critical heat flux is roughly proportional to the square root of the velocity, that is (1)

Clearly this is an approximation since experimental results can be found for which qc is proportional to V" 3 and for which qc is proportional to V 2/3 • For our purposes the representation given by Equation (1) is very attractive since it nicely represents the early work of Gambill and Greene (1958) and the follow-up study by Mayersak et al. (1964). The effect of a shear flow on the boiling process is extremely complex and not yet completely understood. However, it is clear that a shear flow tends to break vapor bubbles away from the heat transfer surface and transport them into the surrounding liquid. This enhances the boiling process and gives rise to an increase in the critical heat flux as indicated by Equation (I). In subcooled boiling, the vapor bubbles

I. The effect of shear and turbulent dispersion caused by a forced flow of the liquid past the heat transfer surface, and 2. The effect of an increased body force caused by an acceleration of the liquid normal to the heat transfer sur-

11

12

A. H. Iversen and S. Whitaker

leave the heat transfer surface and then collapse in the liquid as illustrated in Figure I.

////////////////////////!////

SUBCOOLED

LlQU~

VAPOR BUBBLE \ COLLAPSES

Buoyancy

The influence of buoyancy effects on the nucleate boiling process is clearly evident in the process of "pool boiling" such as one observes when a pan of water is boiled on a stove. In that case the buoyancy force, generated by the difference in the vapor and liquid densities and the gravitatonal force, causes the vapor bubbles to be propelled upwards away from the heat transfer surface. In the absence of a gravitational force, the vapor bubbles would collect at the heat transfer surface and form a vapor film that would terminate the boiling process. As early as 1950, Kutateladze was able to deduce that the critical heat flux during pool boiling was proportional to the fourth root of the acceleration normal to the heat transfer surface:

(2) Here a is the acceleration and g is the standard gravitational acceleration (one "gee"). An alternate proof of Equation (2) was given three years later by Sterman (1953), and the original work of Kutateladze (1950) was extended by Borishanskii (1956) to include the effect of viscosity. Borishanskii's study also confirmed the form of Equation (2). All three of these original studies were based on dimensional analysis, and the first hydrodynamic stability theory by Zuber (1958) confirmed exactly the functional dependence illustrated by Equation (2). The limited experimental data that were available in 1958 were in agreement with

.

c:::::>

GRAVITY

77777777~7777777777777777 NUCLEATING SITE

Figure 1 Bubble Generation and Collapse during Subcooled Boiling.

Zuber's result; however, the existence of a reliable theory generated considerable interest in the influence of acceleration on the pool boiling critical heat flux. Usiskin and Siegel (1961) performed experiments using reduced and zero gravity fields and their results are in agreement with Equation (2). Costello and Adams (1961) carried out experimental studies of the critical heat flux for values of a/ g up to 44, and Ivey (1962) made similar measurements for 1 < (a/g) < 160. In both cases the one-fourth power of a/ g provided a reasonably accurate representation of the experimental data. Between 1961 and 1965 there were numerous experimental studies of this phenomenon and the problem became the object of a special symposium entitled "Effects of Zero Gravity on Dynamics and Heat Transfer" which was held at the Houston AIChE Meeting in 1965. After 1965 the general conclusions put forth by Zuber (1958) were accepted as correct, and the interest in studying the influence of a/ g on the pool boiling critical heat flux greatly diminished. The entire subject was reviewed by Sie-

------------------------------NOMENCLATURE ----------------------------English a:

Ai: A,:

b:

bo:

f g: h: k: L: n:

p: P: (P):

qc: qo: r:

s:

Radial acceleration (cm/s') Interfacial area associated with the averaging volume (cm') Area of entrances and exits associated with the averaging volume (cm') Channel depth (cm) Maximum channel depth (cm) Darcy-Weisbach friction factor Gravitational constant (cm/s') Thickness of the heat transfer section (cm) Thermal conductivity (kcal/cm . K) Length of a single heat transfer section (cm) Unit normal vector Time-averaged pressure (Pa) P + pcp, time-averaged pressure that includes the effect of gravity (Pa) Local volume-averaged pressure (Pa) Critical heat flux (kcal/cm') Uniform heat flux (kcal/cm') Radius of curvature (cm)

T:

To: Tm: v: v:

v: v':

("¥): Vr:

Arclength along a streamline (cm) Temperature (K) Reference temperature (K) Maximum temperature (K) Magnitude of the velocity vector (cm/s) Velocity vector (cm/s) Time-averaged velocity vector (cm/s) ~ - "¥, velocity fluctuation (cm/s) Local volume-averaged velocity (cm/s) Averaging volume (cm')

Greek

t"J.T,Ub: e:

T:

Subcooling (K) bl bo, dimensionless channel thickness Unit vector tangent to a streamline Viscosity (N . s/m') Mass density (kg/m') Wall shear stress (N/m') : Time-averaged viscous and turbulent stress tensor (NI

cp:

Gravitational potential function (cm'/s')

A: p.,: p: To:

m')

13

ULTRAHIGH HEAT FLUX SYSTEMS

gel (1967) and the most recent experimental investigation was carried out by Lienhard and Sun (1970). Their results are in general agreement with Equation (2). Flow Past Curved Surfaces

When a liquid flows past a concave surface at which nucleate boiling is taking place, two phenomena occur. First, the boiling process is enhanced because of the shear generated by the tangential fluid motion (as in the case of linear flow) and second, the radial acceleration, a = vir, gives rise to a buoyancy force which propels the vapor bubbles away from the heat transfer surface as illustrated in Figure 2. This effect was first investigated by Gambill and Greene (1958) who used a swirl flow to achieve a critical heat flux of 17 kW/cm2 (170 MW/m2). This represented the highest known boiling heat flux in 1958 and it clearly established that curved surface flow could be used to generate exceptionally high nucleate boiling heat fluxes. In the work done at Oak Ridge National Laboratory, Gambill and Greene carried out 28 critical heat flux experiments for values of alg ranging from zero to 18,000. Their results indicated that the critical heat flux for a combined shear and buoyancy driven nucleate boiling heat transfer process could be represented as

qc - v" 2(1

+

(3)

(alg)1!4) .

When alg is large compared to one, we have (l

+

(alg)1!4) -

(alg)"4

=

(v2/rg)1!4 -

V

lf2 ,

(4)

and under these circumstances the critical heat flux becomes a linear function of the velocity

(5) It is important to keep in mind that it is not difficult to obtain values of the acceleration that are large compared to g, thus the approximation given by Equation (4) is easily achieved. For example, if the radius of curvature is 2.54 cm, a fluid velocity of 48.8 cm/s leads to a = g. Because of this, concave surfaces can be used to generate critical heat fluxes that are proportional to the velocity rather than the square root of the velocity. This leads to a more efficient heat removal mechanism which in turn permits the use of

lower liquid velocities and the associated lower pressure drops in order to achieve a given heat flux. The original discovery of Gambill and Greene has been used by researchers at Hewlett-Packard Laboratories (Leslie et al. 1983) to develop an enhanced brightness x-ray source. Using a cone-shaped anode with a concave radius of curvature of 2 cm and a fluid velocity of 10,000 cm/s, Leslie et al. were able to obtain a heat flux of 25 kW/cm 2 (250 MW/m2). However, this was not a burnout or critical heat flux and they suggested that their curved surface device was capable of producing heat fluxes two to three times larger than the measured value of 25 kW/cm 2. Although the published experimental results obtained at Hewlett-Packard are rather limited in nature, it is important to note that Leslie et al. interpreted their findings with a critical heat flux correlation identical in form to Equation (5) (linear in the fluid velocity). Since the values of alg were in the range of 10,000 to 50,000, the simplification of Equation (3) to Equation (5) is certainly justified. In 1984, experiments were carried out at Coriolis Corporation by Iversen and Whitaker (1984) using FC 77 as a coolant and a curved channel having a radius of curvature of 2.5 cm. The experimental system is illustrated in Figure 3 where we have shown only the flow channel and the planar heat transfer surface. The heat transfer surface was contained in a vacuum system and the heat flux was provided by an electron beam. A fluid velocity of 1340 cm/s (al g = 726) was used to obtain a "near-burnout" heat flux of 6.3 kW/cm 2 (63 MW/m2). This result can be predicted from the Gambill and Greene (1958) results when a correction is made for the use of a fluorocarbon instead of water. Thus the Iversen-Whitaker experiments represent a direct confirmation of the Gambill and Greene findings. On the basis of the studies of Gambill and Greene (1958) and our own experimental work, we have developed an empirical equation for the critical heat flux for FC 77. This is given by

qc

=

3.6 X 1O- 4v"2 LlTsub(l

+

(alg)lf4) .

......... VAPOR BUBBLE .' COLLAPSES

NUCLEATING SITE

Figure 2 Bubble Generation and Collapse during Subcooled Boiling in a Curved Channel.

lilirrrll Figure 3 Heat Transfer Test System.

(6)

14


where qc IS In kW Icm 2 , v is cm/s, and the subcooling, !lTsub , is in K. When the radial acceleration is negligible Equation (6) reduces to Equation (I), and when alg is large compared to one we recover the form given by Equation

y

(5).

Heat Transfer at a Planar Col/ector The heat transfer section for a large area planar collector is shown in Figure 4. The boundary value problem for heat conduction in this system can be expressed as

a2T

a2T

ay2 + aY?

X

X=O

=

X=U2

(7)

0,

Figure 5

Heat Conduction Domain.

with the following boundary conditions: B.C.l

aT ax

B.C.2

T= To,

B.C.3 B.C.4

=

0,

=

hex) ,

-k-

=

qo,

aT ax

=

0,

aT ay

y

x = 0,

(8) (9)

y= x

=

0, and Ll2 .

(10)

the use of numerical methods; however, the heat conduction process is nearly one-dimensional in the y-direction and under these circumstances we can develop a reasonable approximate solution. We begin by forming the indefinite integral of Equation (7) to obtain

(11)

The region under consideration here is illustrated in Figure 5. In Equation (9) we have used To to represent the boiling point temperature of the fluid flowing past the curved surface. The collector is assumed to be subject to a uniform heat flux which is identified as qo in Equation (10). The boundary conditions given by Equations (8) and (11) represent zero flux conditions that are caused by the symmetry of the system under consideration. An exact solution for the temperature field would require

(12)

The boundary condition given by Equation (I) can be used to express this result as

aT ay

=

_

(qo) _ k

2

J1J=v (a T) dTJ . 1J=0 ax 2

(13)

A second integration leads to BOILING SURFACE

(14)

and we can use B.C. 2 to express this result as

(15)

It is important to note that this is an exact representation

iii i i iii iii

for the surface temperature of the collector; however, this solution is oLlittie value unless we can say something specific about the second derivative, a2Tlax 2. If we neglect the last term in Equation (15) we obtain an upperbound for the surface temperature which we express as

ENERGY FLUX

Figure 4 Heat Transfer Section.

(16)

15


When copper is used to construct the collector and qo is 2 kW/cm2, Equation (16) leads to

To = 200 K, To = 133 K,

hex) hex)

ho = 3.0 mm and = 2.0 mm.

hi

(17a) (17b)

This means that the maximum surface temperature nonuniformity is 67 K over a distance of 0.7 cm which is L/2 in Figure 5. Since heat conduction in the x-direction will reduce this nonuniformity, we would like to estimate the last term in Equation (15). We use the standard estimate

aT ax2 = 2

[aT]

Fluid Distribution-Theory The fluid distribution problem is illustrated in Figures 6 and 7. In Figure 6 we have shown an inlet channel which illustrates that the fluid enters over one-half of the channel and then spreads out to occupy the entire channel before flowing into the heat transfer section. After flowing through the heat transfer section, the fluid passes through the outlet section which occupies the left hand side of the contiguous flow channel. In Figure 7 we have shown a section of the

(18)

Q (L/2)2

FLOW IN

FLOW OUT

in order to express Equation (15) as = T.

TJ \'=0

o[~]

+ qoh(x) + k

0

- (L/2)2

2

h (x). 2

(19)

aT

//

In these equations represents the change in the temperature that occurs in the x-direction and we can estimate this as (20)

TJ

= 1'=0

T. 0

+

qoh(x) k

o - hi)]}. {I + O[h(X)(h (L/2)2

. _'t )

,

Use of this result in Equation (19) leads to

I

(21)

(

\

\

.... _'"

AVERAGING VOLUME

I

I

For the system under consideration we have

h(x)(ho (L/2)2

hi)

=

0(0.05) -

(22)

Since heat conduction in the x-direction will reduce the surface temperature nonuniformity we rewrite Equation (17) as

T(x = 0, y = 0) - To = 190 K , hex) = 3.0 mm and T(x = L/2, Y = 0) - To = 126 K , hex) = 2.0 mm .

' - HEAT TRANSFER SECTION

Figure 6 Side View of Fluid Distribution Channel.

(23a) (23b)

This leads to a surface temperature nonuniformity of 64 K over a distance of 0.7 cm. If ho and are reduced to 2.0 mm and 1.0 mm respectively, Equation (17) takes the form

hi

Tm - To = 133 K, hex) = ho = 2.0 mm and Tm - To = 66.5 K , hex) = = 1.0 mm .

hi

(24a) (24b)

This indicates little change in the surface temperature nonuniformity. If highly uniform surface temperatures are required, they can be achieved using a two-metal construction technique. This technique makes use of two metals having different thermal conductivities in order to compensate for the nonuniform thickness of the planar collector, and the details are described by Iversen and Whitaker (1989). In the next section we describe a fluid distribution system that can be used to achieve the fluid flow illustrated in Figure

4.

t t 't t t t t t t t t ENERGY FLUX

Figure 7 Cross Section of Fluid Distribution Channel and Heat Transfer Channel.


16

inlet flow channel and the heat transfer section. From Figure 7 it is clear that the depth of the fluid distribution channel is a function of position. The fluid mechanical problem is described in terms of the time-averaged continuity equation V'v - - = 0,

Continuity Equation

The method of volume averaging can be illustrated with the continuity equation by forming the average of Equation (25). This leads to

~rV'vdV=O VI JVf- -

(25)

and the time-averaged equations of motion that can be expressed as

v . (pvv)

=

- VP

+

V .

T(T) .

and the use of Equation (31) yields

~

r

V . v dV = s VI JV{- -

(26)

-

In Equation (31) and in Equation (35) the vector 1]. represents the outwardly directed unit normal vector for the averaging volume VI' Since

(27)

+ jv,_ + kv, .

(35) A, -

where is the gravitational potential function. The flow has been assumed to be incompressible and is fully threedimensional: iv,

IY' . (s(y»

+~fn.vdA=O. VI

Here P is a pressure which includes the effect of gravity and is given explicitly by P=p+p,

(34)

'1 .

Y = 0, at Ai

(36)

we' see that Equation (35) reduces to

(28)

Y' . (s(y»

0 .

=

(37)

, The stress tensor T (T) includes both viscous and turbulent = stresses and is given by

On the basis of Equation (32) we can simplify this result to

Y' . (b(y»

(29)

Here PIT represents the turbulent or Reynolds stress. The general solution of Equations (25) and (26) represents a very challenging problem; however, the flow field is constrained by (30)

and this means that the governing equations can be averaged in the z-direction without a significant loss of information. This averaging procedure is best done in terms of the method of volume averaging and the averaging theorem (Leibnitz Rule). This is given by (Whitaker 1989):

~ r Yt/J dV VI JVf

=

S-IY(S(t/J»

+

~VI

f

1].1/1 dA.

(31)

AI

Here VI represents the averaging volume illustrated in Figures 6 and 7, and s represents the volume fraction of the fluid phase which is given by s = b(x,y)lbo

(32)

The interfacial area associated with the averaging volume is designated by Ai' and in our analysis we have in mind the averaging volume will become arbitrarily small so that the average of some function 1/1 defined by (1/1)

= ~ r 1/1 dV vIJV{

becomes a line average in the z-direction.

0,

=

(38)

and here it should be remembered that (y) is a two-dimensional vector. This means that the expanded form of Equation (38) is given by

a

-

ax

(b(v»

a

+ - (b(v» ay·

=

0

(39)

This equation provides a constraint on the flow field illustrated in Figure 6. Momentum Equation

The analysis of the momentum equation begins with Equation (26), and the average of that equation is given by

~

IfVf- -

r V . (pvv)dV = - -VI

vlJvf-

VPdV

+~ VI

r

(40)

V. 1'T) dV .

JVf -

Use of the averaging theorem given by Equation (31) leads to

S-IY' . (sp«yy» +

&f I

-s-'Y(s(P» + (33)

~VI

f

Ar

/+-I fY'·.tldV. VI v,

-

1]. •

pyy dA

1].P dA

(41)


On the basis of Equation (36) we have the simplification

~VIA,J n . pvv-- dA

= 0

(42)

,

and since we have in mind that VI is tending toward zero, we have the simplification

~VI JIJ:? dA = {~J VI A,

IJ: dA}

I

?

A,

(43)

Here we have assumed that the pressure field is symmetric about the centerline of the channel shown in Figure 7. One can prove in a completely general manner that

~VI J. n dA

-E-IV'E

=

A, -

-

(44)

,

and when this result, along with Equations (42) and (43), is used in Equation (41) we obtain

E-

rv . (EP(yy)

=

- V(?)

+

Here Ae represents the area of entrances and exits of the averaging volume shown in Figures 6 and 7. Although both Ae and Ai tend toward zero as VI tends toward zero, one can prove that

~VI J IJ:' A.

AI

E-IV E(?

I - (?) A,

(45)

+~ (

V'. 'fIT) dV . VI JVi - =

Progress at this point becomes very difficult unless we are willing to neglect the variations off and y in the z-direction. This simplification can be expressed as

17

l(T)

dA

«

-

~VI J IJ: • l(T) dA A,

,

(52)

-

and this allows us to write Equation (50) as p(y) . V(y) = - Y(P)

+

1 -V

J

I

Ar

IJ: • l(T) dA .

(53)

-

The last term in this result represents the force that the walls of the channel exert on the flowing fluid. The fluid motion in the channel can be represented by

(n

= (v)~

,

(54)

where (v) represents the magnitude of the averaged velocity vector and ~ represents a unit vector pointing in the direction of the flow. If we a-ssume that the force exerted by the walls on the fluid is parallel to the fluid motion, we arrive at the representation:

~ J. n . VI A, -

'f(T)

=

-~b T O'},

dA =

(55)

1

Here To is the wall shear stress and it is related to the DarcyWeisbach friction factor by (56)

(~~)

where

y

«

(yy) ,

(47)

represents the spatial deviation defined by

y=y-(y).

(48)

It is not difficult to demonstrate that these simplifications are valid when Vb < < 1, although they are likely to be quite satisfactory under less severe conditions. Use of Equations (46) through (48) allows us to write Equation (45) as

In making use of Equation (56) we have assumed that the wall shear stress for flow in a uniform channel can be used to represent the wall shear stress for flow in a nonuniform channel such as we have illustrated in Figure 7. When Equation (55) is used in Equation (53) we have the following fluid mechanical problem Continuity Equation

Y . (b(y)

(57)

0,

(49) Momentum Equation

Further simplification results when Equation (37) is used so that Equation (49) takes the form p(y) . V(y)

= -

V(?)

+

~ JV . ~(T) dV.

p(y) . Y(Y) = - Y(?) -

2 b To~

,

and

(58)

Constitutive Equation (50)

I

TO

The nature of the last term in this result suggests that we avoid the use of the averaging theorem and instead make use of the divergence theorem to obtain

(51 )

=

~G P(y)2)

(59)

The solution of Equations (57) through (59) is much simpler than the solution of Equations (25), (26), and (29); however, we are still faced with a very difficult problem. Streamline Analysis

Considerable information can be extracted from Equations (57) through (59) without solving the boundary value


18

problem associated with the flow illustrated in Figure 6. If we make use of Equation (54) with the left hand side of Equation (58) we obtain (~)

. yW)

(V)1 .

=

yW)

d(~)

P

ds

=

_

V(J5) -

-

270 A . b -

(62)

d(J5)

=

ds ' and

d(v) P(V)1· d;

=

d p(~) . ds (~)

(63)

d

(1 )

= P ds 2: (V)2

Forming the scalar product of Equation (61) with using Equations (62) through (64) leads to

!!.-(! P(V)2) ds2

=

_

d(J5) _ ds

270 .

(64)

1

and

1

-ds ds

-,=0

I

1

=

s=.,.,

(66)

(270) ds. b

-,=0

2: p(v)i

-

1

2: P(v)5

j

(67)

+ S=." (270) - ds. b

s=o

It is not difficult to demonstrate that (V)5