NBSIR 75-807

TURBULENT NATURAL CONVECTION OF LIQUID DEUTERIUM,

HYDROGEN, AND NITROGEN WITHIN ENCLOSED VESSELS

David E. Daney

Cryogenics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

February 1975

Prepared for:

Lawrence Livermore Laboratory

University of California

P. 0. Box 808

Livermore, California 94550

NBSIR 75-807

TURBULENT NATURAL CONVECTION OE LIQUID DEUTERIUM,

HYDROGEN, AND NITROGEN WITHIN ENCLOSED VESSELS

David E. Daney

Cryogenics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

February 1975

Prepared for:

Lawrence Livermore Laboratory

University of California

P. 0. Box 808

Livermore, California 94550

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS Richard W Roberts Director

CONTENTS

Page

1. Introduction 1 1.1 Previous Work 2

2. Experimental 3 2.1 Experimental Approach 3 2.2 Apparatus - General Scheme 6 2.3 Apparatus Details 9 2.4 Test Vessels

2.5 Instrumentat

2.6 Data Reduction 21

2.7 Thermal Properties ..... 23

3. Results 23

4. Discussion 35

4.1 Temperature Profiles 35

4.2 Nusselt Numbers 41

5. Summary and Conclusions 43

6. Acknowledgments 45

7. Nomenclature 46

8. References 48

APPENULX A Heat Capacitv During Cooling 51 APPENDIX B Velocity in the Core of a Vertical Cylinder 53

B.l Quasi-Steady Cooling 53

B.2 Internal Heat Generation 56

APPENDIX C Experimental Results 58

LIST OF FIGURES

Figure Page

2.1 Experimental apparatus

2.2 Complete experimental assembly 10

2.3 Test vessel - 17 1 cm diameter hemisphere 12

2.4 Test ve.-sel - vertical c liuder 13

iii
2.5 Test vessel - sphere 14

2.6 Hemisphere heater 15

2.7 Wall thermocouple installation-copper walls 20

3.1 Summary of Nusselt number data for all cooling

experiments in copper walled vessels. The correlations

and curves are for the LD^ data only 26

3.2 Vertical temperature distributions of the copper walled

hemispheres, sphere, and horizontal cylinder for the

warming experiments. The fluid is liquid deuterium 27

3.3 Vertical cylinders (copper wall) - vertical

temperature distribution for cooling experiments 28

3.4 Nusselt number data for warming experiments in

copper walled vessels 29

3.5 Vertical temperature distributions of the hemispheres,

sphere, and horizontal cylinder for the warming experiments. Unless specified, the fluid is liquid deuterium ... 30

3.6 Vertical cylinders (copper wall) - vertical temperature

distribution for warming experiments 31

3.7 Transient response of LD^ within a copper sphere 33

3.8 Transient response of LD„ within a vertical copper

cylinder (L/D =2) 34

4.1 Dimensionless temperature, T*, within 17.1 cm diameter

hemisphere for cooling experiments. Measurements are in

the vertical plane bisecting the hemisphere 36

4.2 Comparison of the vertical temperature distribution

within the sphere during cooling with the inverted

temperature distribution during warming 40:

C.l Sphere (copper wall) - Nusselt number for cooling

experiments 58

C.2 Sphere (copper wall) - vertical temperature distribution

for cooling experiments 59

C.3 Horizontal cylinder (copper wall) - Nusselt number for

cooling experiments 60

C.4 Horizontal cylinder (copper wall) - vertical temperature

distribution 61

C.5 Copper hemispheres - Nusselt number for cooling experiments . 62

C.6 Copper hemispheres - vertical temperature distribution

for cooling experiments . ... . ... . . . 63

C.7 Stainless steel hemisphere - Nusselt number for

cooling and heater experiments 64

iv
C.8 Plastic-copper hemisphere - Nusselt number for cooling

and heater experiments 65

C.9 Stainless steel and plastic-copper hemisphere - vertical

temperature distribution for cooling experiments 66

CIO Stainless steel and plastic-copper hemispheres with

heaters - vertical temperature distribution 67

C.ll Vertical cylinders (copper wall) - Nusselt numbers for

cooling and heater experiments 68

C.12 Vertical cylinders (copper wall) - vertical temperature

distribution for cooling experiments 69

C.13 Vertical cylinder (copper wall) with heater (L/D = 2)-

vertical temperature distribution 70

C.14 Sphere (copper wall) - vertical temperature profile

for warming experiments 71

C.15 Horizontal cylinder (copper wall) - vertical temperature

distribution for warming experiments 72

C.16 Copper hemisphere (D = 17.1 cm) - vertical temperature

distribution for warming experiments 73

C.17 Vertical cylinders (copper wall) - vertical temperature

distribution for warming experiments 74

LIST OF TABLES

Table Page

2.1 Test vessel characteristics 17

2.2 Thermal properties of the test fluids at 4 atmospheres ... 24

3.1 Summary of Nusselt number correlations for liquid

deuterium cooling experiments 32

C.l Correlation results using the expression Nu = C-Ra

11 .... 75

v
ABSTRACT

Quasi-steady

nitrogen within a cylinder has been

isothermal wall,

numbers

,

fits the deuterium and nitrogen data over the range 7 x 10^ < Ra < 6 x 10

while the hydrogen Nusselt numbers are 8 percent lower. The temperature

field within the vessels is virtually free of horizontal temperature

gradients. A single dimensionless temperature profile characterizes the

vertical temperature distribution for each vessel shape, with the profiles for the sphere, hemisphere, and horizontal cylinder being nearly

identical.

natural convection of liquid deuterium, hydrogen, and

sphere, hemisphere, horizontal cylinder, and vertical

studied experimentally for the case of a nearly

A single expression relating the Nusselt and Rayleigh

0. 352

Nu = 0.104 Ra

Key words: Cryogenic; heat transfer; hemisphere; horizontal cylinder;

liquid deuterium, liquid hydrogen, liquid nitrogen; natural convection;

sphere; vertical cylinder.

vi
TURBULENT NATURAL CONVECTION OF LIQUID DEUTERIUM,

HYDROGEN, AND NITROGEN WITHIN ENCLOSED VESSELS

by

David E. Daney

1. Introduction

Natural convection heat transfer within enclosures occurs in a number of applications. Cooling of turbine blades, heating and cooling

of buildings and liquid storage tanks, cooling of nuclear fuel elements,

and industrial process cooling are only a few examples. Of particular

interest to the cryogenic engineer are the warming and thermal stratification of cryogenic fluids (both liquid and supercritical) in storage

and rocket propellant tanks. Cooling of liquid hydrogen and liquid

deuterium moderators and targets for high energy physics experiments is

another potentially useful cryogenic application of natural convection

heat transfer. However, this latter application has been used only at

very low heating rates because there is insufficient natural convection

heat transfer design information. In order to provide such information

the experimental investigation reported here was undertaken.

The three parameters of primary interest in the design of these

vessels are the average heat transfer coefficient, the average bulk

fluid temperature, and the maximum fluid temperature. The bulk temperature gives average fluid density and, hence, the mass of fluid

within a vessel. The maximum temperature gives the proximity to the

boiling point, which in some applications is to be avoided. Also of

considerable theoretical interest is the temperature field within a vessel because of the information it contains about the flow patterns.

In this study the average heat transfer coefficient and the temperature

field were determined for turbulent natural convection heat transfer

with liquid deuterium (LD2) , liquid hydrogen (LH^) , and liquid nitrogen

(LN„) enclosed within spheres, hemispheres, horizontal cylinders, and
vertical cylinders with nearly uniform wall temperatures. Quasi-steady

cooling (that is, cooling with a constant wall to fluid temperature

difference) was the primary mode studied, although results are also

given for quasi-steady heating, transient cooling, and heating with

spatially distributed electrical heaters. The Rayleigh numbers ranged

8 11 from 7 x 10 to 6 x 10 and the temperatures range from 20 K to 26 K

for LH

2 and LT> 2 and 78 to 88 K for LN2>

1.1 Previous Work

A considerable body of literature exists on the subject of natural

convection within enclosures, but it is concerned primarily with laminar

free convection, or with transient free convection. None of this previous

work addresses either the quasi-steady turbulent convection considered

here or the fluids used in this study. Although numerical solution of

the governing two-dimensional partial differential equations has been

successful for laminar flow, computation times have generally been on

fl 2] the order of an hour on an IBM 7090. ' At high Rayleigh numbers

numerical solutions become impractical because the thin boundary layer

[31 flows which have been observed experimentally would require such a fine grid spacing near the container walls that computation times would

become excessive. Approximate analysis using the integral method has

[4] been successively applied to open thermosyphons , narrow tubes with

internal heat generation [ ^^ and transient convection in vertical cylin-

[3] ders with uniform wall heating; but it apparently has not been

successively applied to other enclosure geometries or to the uniform wall

temperature case.

Of the previous experimental studies of natural convection within

enclosures, Schmidt^"' gives Nu vs. Ra and some temperature profile data

for transient turbulent free convection in spheres up to 50 cm in diameter

using water and various alcohols. Evans and Stefany[^ and Maas^"'

investigated transient laminar free convection in horizontal cylinders

using water, n-butanol, methanol and water-glycerine mixtures. Deaver

[91 and Eckert studied quasi-steady laminar free convection within horizontal cylinders using water and ethylene glycol. A considerable effort,
both analytical and experimental, has been spent on transient free

convection in vertical cylinders and on the associated thermal stratification. Clark has summarized this work; and some of the more relevant experimental studies are by Evans and Stefany!^ Evans, Reid and

DrakeJ

3

^ Schwind and Vliet[ 11 -' and Tat om and Carlson.

2. Experimental

2.1 Experimental Approach

Selection of the quasi-steady cooling method with constant wall-tofluid temperature difference offers a number of advantages. The steady

state is more amenable to analysis than the transient state, so steady

state experimental data are more easily compared with theory. Because

simple, well-defined boundary conditions result, automatic data acquisition and reduction are easily accomplished. Finally, quasi-steady

cooling simulates steady state free convection with uniform internal

heat generation — an important consideration for cold neutron moderator

experiments

.

This analogy between quasi-steady cooling and uniform internal heat

generation results because of the similar effects on the convection

process caused by the fluid heat capacity and by uniform internal heat

generation. Inspection of the energy equation shows this analogy is

exact. For a low Mach number, constant pressure, and constant thermal

conductivity system, the energy equation is

pC [-^ + W • VT] = k V

2 T + q . (2.1)

p dt V

Rearranging equation (2.1) so that the terms defining the velocity and

temperature field are on the right hand side gives

pC |A - q = k V T - pC W • VT. (2.2)

p 9t v p

The individual contributions that the terms on the left hand side of

e 4 uation (2.2) make to the velocity and temperature fields are indistinguishable since the variables are separable in this fashion. Thus, the
velocity and temperature fields within a vessel for the constant rate of

cooling case,

and

pC t— = Constant

p dt

q = 0, v should be the same as for the steady state uniform internal heating

case

,

q = Constant

^v

and

PC | = 0.

p dt

More formally, by transforming equation (2.2) using (J) = T - T , the w

requirements for quasi-steady cooling are:

9Tw

pC -tt— = Constant

p dt

and

T - T = Constant, w

In controlling the experimental conditions the latter criterion was used,

whereas in the selection of the data to be analyzed, both criteria were

used.

In selecting correlating parameters, we assumed that steady-state

parameters are also valid for quasi-steady conditions - an assumption that

is supported by these experimental results. Thus we use the expression

Nu = C Ra

U

1 to correlate the heat transfer coefficient data. The characteristic

temperature difference used in the Nusselt number and Rayleigh number is

1^ - T^ and, the fluid properties were evaluated at the bulk fluid

temperature T, . We use the surface average heat transfer coefficient, so

b that the Nusselt number is given by
Nu = — =

( q /A) °

. k

(T - T )k

b w

The characteristic length is the vessel diameter for the sphere, hemispheres, and horizontal cylinder; and the height for the vertical cylinders.

The temperature profile data were correlated using the dimensionless

temperature

T - T

T* = ^ , T - T

b w

which is simply the local temperature difference, T - T , normalized by the w

characteristic temperature difference, T - T . b w

The parameter T* yields a single dimensionless temperature profile

for each shape when the flow is fully developed, because identical T*

profiles imply similitude of velocity fields. This connection between the

velocity and temperature fields is most easily visualized by considering

the uniform internal heat generation case. If we follow a fluid element

rising within the vessel, then, (in the absence of heat transfer or mixing) the temperature rise of that element is proportional to the time

of travel from our datum point, and the slope of the temperature, dT/dx,

is inversely proportional to the velocity. Changing the heating rate or the magnitude of the velocity will change the temperature difference at

a given point within the vessel, but it will not change the temperature

difference relative to the volume average (bulk) temperature difference

within the vessel. For instance, in the simple one-dimensional case of a fluid with internal heat generation rising at constant velocity within an insulated vertical cylinder, the temperature rise at the mid-point will

always be one-half the temperature rise at the top, and the temperature

rise at 1/4 the height will be 1/4 the total temperature rise regardless

of the heating rate or cylinder height. Heat transfer and mixing complicate the picture, but as these results demonstrate, the T* profile

within a given geometry is constant for fully developed turbulent flow.

5
From the definition of T* , we find that at the wall

and for the bulk fluid

T* = 0 , w

T* El.

b

We obtain the actual temperature at any point within a vessel using the

identity

T - T E T* (T -. T ) , w b w

where T - T is obtained from the Nusselt-Rayleigh number correlation. w

2.2 Apparatus - General Scheme

Figure 2.1 is a schematic of the experimental apparatus. The

different test fluids used in the experimental vessels required different

bath fluids: a liquid hydrogen bath for the liquid deuterium and

hydrogen tests, and a liquid nitrogen bath for the liquid nitrogen

tests. Consider a liquid hydrogen experiment. The hydrogen bath is

warmed to about 28 K by bubbling warm hydrogen gas through the liquid

and pressurizing the dewar to about 7.5 atm. After the fluid in the

test vessel has come to within 0.5 K or less of the bath temperature,

the valve to the vent line is cracked, and the temperature of the bath

falls as the bath pressure is bled down to atmospheric pressure or below. During the process a constant temperature difference is maintained between the fluid in the test vessel and the vessel wall by

observing the signal from one of the differential thermocouples and

adjusting the valve-opening accordingly. If the fluid properties were

independent of temperature, cooling with a constant rate of heat removal

would result. In practice, however, variation of the thermal properties

with temperature results in a slowly decreasing heat transfer rate.

6
H 2 Gas

Vent

Dewar

Ballast Volume

(concentric tube

capacitor

)

Platinum

Resistance

Thermometer

Thermocouple

Reference

Junction

" Support Tube

3

Liquid H 2 Bath

Liquid H 2 or D 2

Differential

Thermocouple

Test Vessel

xWarm - up Line

Figure 2.1. Experimental apparatus,

7
During the cooling (or heating) process, temperatures within the

test vessel, and at the test vessel wall were measured with differential

thermocouples referenced to the bath temperature, and the absolute

temperature of the bath was measured with a fast response platinum

resistance thermometer. From these measured temperatures the quantities

of interest may be calculated: T , T_, . , - T , T - T , and 3T^/9t w fluid w b w b

(which gives q) . The bulk temperature, T, , and the average wall temb

perature are obtained from weighted averages of the fluid and wall

temperature measurements.

Most of the data were collected during the cooldown mode of operation. The constant-temperature-difference operation was easier to

achieve in this mode, and the bath was uniform providing a good reference temperature for the thermocouples. Data were also collected

during warm-up, although the experiment was more difficult to control,

and the accuracy was lower.

Some experiments were also conducted using spatially distributed

heaters (concentric coils of 1.5 mm-diameter heater wire) to simulate

internal heat generation. Because the quasi-steady experiments require

special equipment and are somewhat difficult to perform, these much

simpler heater experiments were made to determine if they would yield

results similar to the quasi-steady experiments. If similar results

were obtained, then complex systems involving several modes of heat

transfer might be simulated using heaters in those places where internal

heat generation occurs.

8
2.3 Apparatus Details

Figure 2.2 is a schematic of the complete experimental assembly.

The 46 cm-diameter by 165 cm-deep test dewar is multilayer-insulated and

has a 6.7 atm gauge design pressure. It is flanged at the top and may

be unbolted from the top plate and lowered into a pit in order to gain

access to the experimental vessel and instrumentation.

An airmotor driven hoist can be used to raise the experimental

vessel into the vapor space above the liquid bath. This feature makes

it possible to warm the experimental vessel above the bath temperature,

allowing a greater temperature span to be traversed during an experiment. However, it was found to be too difficult to control the temperature of the experimental vessel in this mode of operation, so the

practice was abandoned after the first two series of tests.

The ejector pump is used to pump the bath below atmospheric pressure

(0.82 atm in Boulder), thus reducing the minimum attainable bath temperature to 15 K for hydrogen. This extension of the operating range is

particularly useful at the higher cool-down rates because it allows more

time to achieve quasi-steady conditions. High pressure nitrogen gas is

used to power the ejector since safety considerations require that there

be no contamination of the hydrogen by air within the apparatus and its

associated piping.

The use of 150 liters of liquid hydrogen at pressures ranging from

plus 6.7 atm to minus 0.7 atm gauge in the test dewar presents a serious

potential hazard which required careful consideration. The test dewar

and main control panel were placed in a room (6 meters high) with ex- plosion-proof electrical fixtures and an exhaust fan on the roof, which

changed the air in the room once every two minutes. All electronic

equipment was located in a separate room with an auxiliary air-purged

control panel for operation of the experiment. Thus, the personnel were

isolated from the dewar when it was pressurized.

9
C\J

crc^cr

to o

UJh- tr<

Q. -1

—d&—:

I*

§ "O uj < cr > Q_X

X

Z>Q-.-f <CL CD > a

CM

CM

X

5&S_2.

CM z

cr _i

CE 3

-

UJ Fi CO

UJ _ O

cr o y

X >j Q Q_

IQUI SUP 8

toCM

0)W

CO

CO

4-1 C

C1J 6

•H

01a

x

0)

0)

4-1 a> H

a

eou

(1) U

o

•rl fa

10
The valves with the A prefix are located on the main control panel

adjacent to the test dewar, and the B prefix valves are on the auxiliarypanel in the instrument and control room. Provision has been made for

evacuating and purging both the dewar and the test vessel with nitrogen

and hydrogen gas. The bath is warmed by bleeding in gas via valve B-4,

and cooled by bleeding down the dewar pressure via remotely operated

valve B-15.

It was found during the first experiments that a gradual loss of

liquid from the ballast volume (see figure 2.1) occurred each time the

experimental vessel was temperature cycled because gas was vented through

the relief valve when the experimental bath was warmed. In order to

prevent this loss, the high pressure gas holder was added to the system.

A standard industrial air cylinder with a 10 cm-bore by 61 cm stroke was

adapted for this purpose. By maintaining a constant pressure (4 atm

absolute) on the top side of the piston, the test vessel pressure was

held constant during each test.

2.4 Test Vessels

The design of the test vessels (see figures 2.3 through 2.5) was governed largely by the test environment of differential pressures up to

3.4 atm in either direction, and temperatures down to 15 K. Wood's

metal solder provided a hermetic seal of the final assembly joint, and

an alumina-filled epoxy sealed the thermocouple leads. Wires with

enamel and polyimide insulations gave a leak- tight bond with the epoxy.

3 Nearly adiabatic walls were provided by high density (0.1 gm/cm to

prevent crushing by the pressure) styrofoam or glass reinforced epoxy.

Because of the high coefficient of thermal expansion of liquid

hydrogen and liquid deuterium (about two percent per degree Kelvin) , a liquid ballast volume (see figure 2.3) must be connected to the test

vessel to assure that the vessel remains full at all times. To prevent

boiling or vapor formation in the test vessel, the surface of the liquid

in this ballast volume is pressurized to about 4 atm with helium gas.

No evidence of helium gas desorption in the test vessel was observed.

11
Fill line

3

Support tube

1.27 cm .wood's Metal — Back plate,

304 stainless

Support rod

Styrofoam cap

Wall

thermocouple

1.27 cm-4* h

17. 1 cm I.D. Hemisphere,

.077 cm wall stainless or .320 cm wall ETP copper

Epoxy

lead-thru

" Styrofoam

insulation

Ballast volume and concentric

capacitor liquid level sensor.

Figure 2.3. Test vessel - 17.1 cm diameter hemisphere

12
/- High density

/«/ Styrofoam

/ //////////////

34,3 cm

29.5 cm

Brass

Soft solder

7.937 cm O.D.

Copper tube

.24 cm O.D.

Stainless steel

Thermocouple

Support

--Wood's Metal

r Silver solder

Brass

Nylon Cup

Figure 2.4. Test vessel - vertical cylinder,

13
Figure 2.5. Test vessel - sphere.

14

Several experiments were run using a spatially distributed heater

to simulate internal heat generation, as discussed in section 2.2. The

heater used in the hemispheres (figure 2.6) has three concentric helical

coils wound from a single length of heater element. This element has

two 30-gauge resistance wires embedded in ceramic powder, and clad in a 1.5 mm-diameter thin walled sheath of stainless steel. Braces soldered

to the element give it the structural rigidity required to hold its

shape. The heater element used in the vertical cylinder was simply a 5-cm diameter vertical helix, 10 turns long. Both heaters have sufficient surface area to prevent boiling at the heat fluxes achieved in the

experiment.

In order to study the effects of the wall resistances on the

convection process, three hemispheres with different wall resistances

were tested:

1) a 3 mm-wall copper hemisphere with a negligible resistance

across the wall and a resistance across the outside boiling

boundary layer which was determined experimentally to be

T - T. = 3.9 (q/A) w bath w

2) a 0.76 mm-wall stainless steel hemisphere with a calculated

total resistance (wall plus outside film) of

T - T, , = 7.7 (q/A) w bath H w

3) a plastic-copper laminated hemisphere having a 0.05 mm-thick

layer of polyparaxylyene vapor deposited on the inside, and

having a thermal resistance given by

T - T, = 10.0 (q/A) . w bath w

A summary of the experimental vessel geometries and characteristics

is given in Table 2.1.

16
17
2.5 Instrumentation

Chromel vs. gold - 0.07 atomic percent iron differential thermocouples were chosen to measure the enclosed fluid and vessel wall temperatures because of the high sensitivity these thermocouples exhibit at

liquid hydrogen temperatures (approximately 16 [J.V/K) . The wire diameters

are 0.15 mm for the gold-iron and 0.22 mm for the chromel, including the

.01 mm thick enamel insulation. The reference junctions lie in the

liquid bath as illustrated in figure 2.1. Thirty-six gauge thermocouplegrade copper wires traverse the distance from the bath to outside the

dewar top plate, and the 20-meter long copper leads from the dewar to

the instrument room have both electrostatic shielding (braided copper)

and electro-magnetic shielding (iron conduit)

.

Thermal tempering of the thermocouples within the test vessels was accomplished by simply cantilevering the wires horizontally for about 4 cm from their support rod as shown in figure 2.3. Two factors make this

technique suitable. First, except for a thin boundary layer near the

wall, temperature gradients were found only in the vertical direction.

Second, from temperature profile data collected in the vertical cylinder

experiments, core velocities are estimated to be on the order of 1 cm/sec.

Thus the horizontal thermocouples traverse an isothermal region where

they are subjected to cross-flow velocities on the order of 1 cm/sec.

Under these conditions the thermal tempering error is calculated to be

-3

less than 10 K.

The thermocouple junctions which measure the fluid temperature are generally aligned on the vertical axis in the sphere, vertical cylinders,

and horizontal cylinders; and aligned along a vertical line displaced

from the back plate, 1.0 cm for the 17.1 cm-diameter hemispheres, and

0.5 cm for the 6.5 cm diameter hemisphere. In a few tests some of the

thermocouple junctions were distributed horizontally in an attempt to

measure horizontal temperature gradients.

Measurement of the wall temperature has often presented something

of a problem in heat transfer measurements, and the most common technique

18
that of inserting thermocouples in long, small diameter holes parallel

to the wall surface—was not practical here. Both the two-dimensionally

curved wall surfaces and the desire to keep the wall as thin as structurally possible (in order to reduce the wall's heat capacity) dictated

that another technique be chosen. As illustrated in figure 2.7, the

thermocouple is wound in a spiral for thermal tempering. The polyimide

film, which is epoxied to the outside of the wall, provides electrical

insulation from the wall, and the styrofoam cap provides thermal insulation from the bath. This technique is suitable only for walls with a high thermal conductance if the wall temperature drop from the edge to

the center of the cap is to be kept small. Thus the inside wall temperature for the stainless steel and copper-plastic walled vessels was obtained using the outside film resistance measured for the copper

vessels and a calculated wall resistance. The thermal resistance across

the polyimide film is the principal source of wall temperature error —

amounting to about one percent of T, - T . The measured average wallb w

to-bath temperature difference (given in section 2.4 as T - T, , = w bath

3.9 (q/A)) is in good agreement with previously measured boiling liquid

hydrogen temperature differences.^"^

The bath temperature was measured by a small (5 mm square by 1 mm

thick), fast-response, ceramically encapsulated, platinum-resistance

thermometer (PRT) , which sits in the midst of the thermocouple reference

junctions as indicated in figure 2.1. The systematic error in the PRT

calibration is believed to be 0.05 K or less — a more than adequate

accuracy since the bath temperature is needed primarily for the determination of the thermal properties.

The microvolt level thermocouple signals were amplified by high

quality, wide band, chopper stabilized, differential DC amplifiers. A

16 channel data logger unit (consisting of a digital volt meter, a scanner, a digital clock, and a magnetic tape drive) recorded the thermocouple and PRT outputs on magnetic tape at a rate of one scan per second.-

The rms noise of the complete data acquisition system is about 1/2

microvolt (equivalent to .03 K) and the drift of the system between zero

19
INTERIOR of eP<> x y

of VESSEL

Figure 2.7, Wall thermocouple installation - copper walls.

20
readings is about 1/3 microvolt referred to the input. In one experiment,

two thermocouples were imbedded in a copper block which was heated above

the hydrogen bath temperature by as much as 8 K, and the agreement was generally better than 0.05 K. Dip tests of the thermocouples in liquid

nitrogen and liquid hydrogen indicated a maximum discrepancy of 1.2%

between the observed emf and the thermocouple reference table used

u [13] here.

To summarize the error discussion, the agreement between thermocouples was generally better than 0.05 K, and the estimated systematic

error was 3 percent or less of T - T . The systematic error in the heat

w

flux is estimated to be less than 10 percent for the insulated wall

vessels (all vessels except the sphere) and about 5 percent for the

sphere. The overall systematic error of the correlations is estimated

to be less than 13 percent for all of the fluids used in this study.

For other fluids, an additional systematic error of 10 percent could be

present because of the uncertainty in the thermal properties (principally

the thermal conductivity) of liquid deuterium. The random error in

Nusselt number for the overall correlation is typified by a standard

deviation of 11 percent.

The liquid level in the ballast volume was determined from the

capacitance of this concentric tube device (see figure 2.3). It was originally hoped that an accurate determination of the bulk temperature

in the experimental vessel could be made from this liquid level measurement (in a manner analogous to mercury-glass thermometry) but the practical

calibration problems involved were too time consuming for the variable

density, variable dielectric constant fluids. However, this liquid

level sensor was a useful aid in the operation of the experiment.

2.6 Data Reduction

Data reduction is a two-step process. First, the emf s are converted

to temperatures and heating rates by a digital computer. Then, these

temperature and heating rate data are scanned by eye, and points sufficiently quasi-steady in nature are selected for evaluation of the

Nusselt number, Rayleigh number, and dimensionless temperature profile.

21
The bulk temperature of the fluid is calculated from a weighted

average of the interior thermocouple readings. Weights are assigned as the fraction of the total volume occupied by a horizontal segment having

the thermocouple located at its vertical center. An absence of horizontal

temperature gradients in the bulk of the fluid makes this the appropriate

method of weighting. The boundary layer is neglected in this calculation, but its contribution should be small, since attempts to directly

measure the inside wall temperature indicated that the boundary layer is

only a fraction of a millimeter thick.

The average wall temperature is calculated from a weighted average

in a manner similar to the bulk temperature calculation. Weights are assigned according to the fraction of the total wall surface area between

horizontal planes equidistant from the thermocouples.

Because the inside wall temperature of the stainless steel and the

plastic-copper hemispheres could not be measured, these wall temperatures were calculated using the wall and film resistances given in

table 2.1.

The heat transferred is determined from the time rate of change of

the bulk temperature according to the expression

9T

q - PVC ^- .

For warm-up type experiments, in which fluid is expelled from the vessel,

C is simply the specific heat at constant pressure, C . For cool-down

experiments, however, allowance must be made for fluid entering the

vessel cooler than the fluid within the vessel. In this case, as shown

in appendix A,

C = C + B (i. - i)

P J

where i_. is the specific enthalpy of the fluid entering the vessel, and

i is the specific enthalpy of the fluid within the vessel.

22
In evaluating the Nusselt and Rayleigh numbers, the characteristic

temperature difference is T - T , and the fluid properties are evaluated

D W

at the bulk fluid temperature, T . The characteristic length is the

vessel diameter for the spheres, hemispheres, and horizontal cylinders;

and the height for the vertical cylinders.

2.7 Thermal Properties

Table 2.2 lists the thermal properties of the test fluids as a function of temperature. Because of the large relative span in absolute

temperature covered by deuterium and hydrogen, the properties of these

two fluids exhibit a particularly strong temperature dependence.

A large uncertainty is assigned to the thermal conductivity of

liquid deuterium because of the discrepancy (20 percent) between the

F 2 4 1 T251 conductivity values given by Powers et al., and Roder and Diller

for liquid hydrogen. The Roder and Diller values for hydrogen are believed to be correct, and since Powers et al. used the same apparatus

for and LR^ , the Powers values for have been reduced by 20

percent. In spite of this correction a 20 percent uncertainty is

possible in the thermal conductivity. As will be discussed in

section 4.2, however, intercomparison of the Ll^ and LD^ Nusselt numbers

indicates that this uncertainty may be closer to 10 percent.

3. Results

Figures 3.1 through 3.6 and table 3.1 summarize the results for the

copper walled vessels. Appendix C gives the complete results for each

vessel shape, wall material, and type of experiment. The curves and

expressions for Nu given on the graphs were obtained from a least squares

curve fit to the expression Nu = C^Ra11

. Both floating exponents and

fixed fractional exponents were tried in these curve fits, and when the

increase in the standard deviation was less than 5 percent of the

standard deviation for the floating exponent [i.e., z (fractional

exponent) < 1.05 Z (floating exponent)], the more convenient fractional

exponents are given. Only the liquid deuterium and liquid nitrogen data

were used to obtain the expressions and curves shown in the figures

23
ft

<3* o o co co o ^ tn n pj M H H H NMNfJNHH

W

<D

S-J 0) s:W

o

eu

CO

CXco

ea. r—h 60

co in * _

ini co in r-

*

oo co i—i r- co o

o co r- (Ni %o cni oo m n ^ in in

o

O(Nl O t- O (Nl (Nl o o o o

O O i-H rH i-H

oo xD ro o r— ^* t—i co co co co csi rj cm

CO

eno

i-i

cw

0) •HU

(-1 U

QJ |

P,on

(ft

ale CO u E

01 u

Q_ H "oE CN

H

W

Q

P

3

a

OQ COO O H , o <—i r- <ni co co csj (Ni ro

oo oo o o oo

ini r- a- ro „ r-i

(Nl ini ro (ni co ^>

2

w

cG

cEQ

X

a

i o

r-1 ^J" a- co co O vD CO r—I O

(Nl i-H i—< r—i f—

I

oo r- oo vO ,

i o\° o ^

2

W

a

o

CEH

a o in in in o c

r- sO T o in in x. in -r o Csj o o\° r- sO in -T ro

~r CO in CO CO co CO ro O o o o S3 o

o O o o O

o o o c o o o o o O

f-H ^ r—* t~~ *f a^cr-oococor-r-, , o\° NNrjcgcocjNtM in o o o o o o o rsi o o o o o o o

,0H

O O O M O

n (n in co n i rH (NI O O O O O ,_, o o o o o

o n %t vo co m c

(Nl(N](Nl(NI(NlCE[Jp

r— -3" 'S* oo h (*i in co N i o o o o o

o o o o o

XI

0)

rt

. E

vD 00 O (Nl «3« f m

r—I .—I (Nl (Nl (NI 03 U

m m o (M t r~

in in in in o o , , OOOOOOO^j o o o o o o CN o o o o o o o

. E NM^acococoKWD <f OCOOMVvO •> to _

.5 rt

0)u

C

24
because the hydrogen data have a mean negative deviation of 8.4 percent

from the deuterium and nitrogen data. The standard deviation of the curve

0 352 through the cooling data for LD^ in copper vessels, Nu = 0.104 Ra ' , is 10.4 percent.

The warming data (figure 3.4), although of lower accuracy than the

cooling data, demonstrate that, as expected, the Nusselt numbers are essentially the same for heating and cooling. The 7.2 percent systematic

difference between the heating and cooling data is within the experimental error.

Use of the dimensionless temperature,

T - T

T " - — , T, — T

b w

results in a single dimensionless temperature profile for each shape

when the flow is fully developed. Because of the virtual absence of

horizontal temperature gradients within the vessels (except in the

boundary layer) , the temperature in any given horizontal plane is

characterized by a single temperature. The standard deviation of T* is

about 0.03 at most points within the vessels. At the top of the vertical

cylinders, a of T* is as large as 0.08.

Wall temperature profile data are omitted for the stainless steel

and plastic-copper hemispheres (Appendix C) because no direct measurements were made. For these vessels, only an average wall temperature was calculated as discussed in section 2.6. A single wall thermocouple was mounted (at the horizontal pole) on the 6.5 cm diameter hemisphere

because of this vessel's small size. Generally larger wall temperature

gradients occur in the warming experiments (comparing figure C.6 with

figure C.16 and figure C.12 with figure C.17) because of the lower

outside wall heat transfer coefficients in these experiments.

Following the quasi-steady cooling period, transient cooling of

the vessels occurred with a nearly constant bath temperature. Figures

25
t

1 1— fTn—i—i—i

1 1 1 n i

26
Figure 3.2. Vertical temperature distributions of the copper walled

hemispheres, sphere, and horizontal cylinder for the warming

experiments. The fluid is liquid deuterium.

27
28
29
-0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

T*

Figure 3.5. Vertical temperature distributions of the hemispheres, sphere,

and horizontal cylinder for the warming experiments. Unless

specified, the fluid is liquid deuterium.

30
L/D

-0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

T*

Figure 3.6. Vertical cylinders (copper wall) - vertical temperature

distribution for warming experiments.

31
Table 3.1. Summary of Nusselt number correlations for liquid

deuterium cooling experiments.

Nu = C, Ra11

Test Vessel n

All Shapes 0.104 0.352

" » 0.166 1/3

Sphere 0.446 0.284

Hemispheres (Copper) 0.164 1/3

Horizontal Cylinder 0.151 1/3

Vertical Cylinders 0.171 1/3

32
33
O CO V£5 ^ CVJ O

nvii) iNi^nvM

i _)iina

1)

nvMJL _>nna

1

34
3.7 and 3.8 show cooling curves for two such transient experiments.

Both calculated curves are based on the quasi-steady cooling correlations, but the solid curve assumes the heat transfer coefficient depends

on the instantaneous value of T. - T , while the dashed curve assumes a

b w

constant heat transfer coefficient that is evaluated using (T, - T ) . . . b w mitia

The constant h curve shows better agreement with the experimental data.

This is in accord with the results of Evans and Stefanyl''"' who observed

constant h cooling in their transient experiments.

4. Discussion

4.1 Temperature Profiles

The temperature field within a vessel provides considerable information about the convective process. Figure 4.1, which is a composite

of the temperatures measured in several different hemisphere experiments,

indicates the virtual absence of horizontal temperature gradients within

the bulk of the fluids. Less extensive measurements in other vessels

gave similar results.

A comparison of the temperature profiles for the sphere, hemisphere,

and horizontal cylinder, figure 3.2 indicates that a single dimensionless

temperature profile would give a reasonable description of the temperature

field within any of these vessels. This is not too surprising considering

the similarity of the shapes.

We obtained information on the boundary layer thickness while

attempting to measure the inside wall temperature directly. Thermocouples 0.1-mm thick were glued to the inside wall of the stainless

steel hemisphere. These thermocouples measured temperatures about

midway between the fluid temperature and the calculated wall temperature,

indicating a boundary layer thickness of only a few tenths of a millimeter. This thickness agrees with the value of 1-mm or less observed by

[31

Evans et al . Thus emerges a picture of a thin boundary layer flow

down the walls, and a slowly rising core free of horizontal temperature

gradients for the cooling experiments. For the warming experiments the

flow directions are reversed.

35
Figure 4.1 Dimensionless temperature, T*, within 17.1 cm diameter

hemisphere for cooling experiments. Measurements are in the

vertical plane bisecting the hemisphere.

36
The relatively small vertical temperature gradient in the upper

portion of the sphere, hemispheres, and horizontal cylinder for the

cooling experiments (lower portion of the vessel for heating experir 6

1

ments) indicates highly turbulent mixing in this region. Schmidt

observed this condition in Schlieren photographs of free convection

within a horizontal cylinder. The temperature gradient for cooling in

the lower portion of the sphere, hemispheres, and horizontal cylinder

indicates that boundary layer fluid enters the core all along this

region. If we consider the uniform heat generation analogy, then

potential flow from a source at the bottom of the vessel would give _—__ a, __ r\, — Area normal to flow

d(x/L) dx w

as shown in the Appendix B. Thus the temperature gradient should in- crease proceeding from the bottom to the middle. However, the temperature profiles shown here exhibit the opposite behavior, indicating a substantial deviation from this type flow.

r 6

1

Schmidt measured the vertical temperature profile within spheres

during transient heating. His profiles show a maximum temperature

difference at the mid-point followed by a small decrease in the AT - proceeding towards the bottom. In contrast, our heating profiles are nearly isothermal in the lower portion as shown in figure 3.5.

The absence of wall temperature data in Schmidt's paper prevents a more quantitative comparison.

The difference between the temperature profiles for the small and

large copper hemispheres (figure 3.2) apparently results because the

tests with the smaller hemisphere covered the region of transition from

laminar to turbulent flow. The Rayleigh number for these tests ranges

8 9 from 7 (10) to 6 (10) , spanning the approximate transition Rayleigh

number of 10 . At the lower Rayleigh numbers, the T* ' s in the upper

portion of the hemisphere were somewhat larger than shown in figure 3.2,

while at higher Rayleigh numbers the T* profile nearly coincides with

that of the larger hemisphere, indicating that the turbulence had become

nearly fully developed.

37
The linear portion of the vertical cylinder temperature profile

(figure 3.3) suggests a flow model (similar to that adopted by Evans,

[3] Reid and Drake ) consisting of a thin boundary layer at the wall and

vertical slug flow of the core. Using this flow model it is possible to

predict the core velocity, w^, if 9T*/9(x/L) and Nu = f(Ra) are known.

The relationship is probably most easily understood using the internal

heat generation approach, although identical results are also obtained

for the quasi-steady case. As discussed previously, a heat generating

fluid creates a temperature gradient in the core as it rises. In the

absence of heat transfer or mixing, this temperature gradient will be

linear if the core velocity is constant. Noting that

8T =

1 dT

Bx w 9t

c (4.1)

and that

pC

P aT " v (A. 2)

it is shown in Appendix B that the core velocity is given by

= iM^> (4.3)

C PC D(|T

-)

r p OX

or if Nu = Q Ra 1/3, then

m 4(L/D)(q/A) 1/4(4.4)

c PCp C

^

p

L

2 9(x/L)

2 For (q/A) =0.1 W/cm the resulting core velocity in these experiments

is 0.5 cm/s.

The 1/D dependence of w^ in equation 4.4 is to be expected from the

continuity equation since the core cross sectional area is proportional

2 to D , while the boundary layer area is proportional to D. The dependence, w ^ L, indicates that the volume flow rate in the core, and

c therefore in the boundary layer, is proportional to the cylinder length

L. This result compares to

3/4 Volume flow rate ^ L

38
for free convective laminar flow where Nu <x Ra"^^,^^ and to

XT 1 £1 nr T l- 2 Volume rlow rate — L

0. 4 tor turbulent flow where Nu <x Ra , according to integral equation

[271 analysis for an isothermal, vertical flat plate. The source of

this w L dependence is the observed independence of 8T*/9(x/L) on

L. Because the cylinder height, L, was only changed by a factor of 2, a weak dependence of 9T*/9(x/L) on L would go undetected.

In free convection, heating and cooling are inverses, and inverting

either the direction of gravity (coordinate system) or the sign of the

temperature difference should have the same effect. Inverting both

the coordinate system and the sign of the temperature difference should

leave the temperature field unchanged. Figure 4.2 compares the temperature profile within the sphere during cooling with the inverted sphere

heating temperature profile, where the inversion is made using the

expression

(x/D) . - 1 - (x/D) , inverted

i.e., the coordinate system is simply flipped upside down. As might be

expected, the agreement between the two sets of data is excellent. The

agreement for the copper hemispheres is equally good, but the slope of

the vertical cylinder heating curve is slightly higher than the cooling

data - probably reflecting the effect of the greater wall temperature

gradient. Comparing the horizontal cylinder temperature profiles, the

poorer agreement is primarily the result of distortion caused by the

slightly negative T* observed in the top of the cylinder during heating.

The temperature profiles for the stainless steel and copper-plastic

hemispheres (figure C.9) are in close agreement with the copper hemisphere profile (figure C.6), indicating no substantial changes in the

convection process for the increased wall resistances studied here.

The slight increase in the maximum temperature with increased wall

39
1.0

0.8

0.6

X/D

0.2

I 1 1 1 I I 1

1 1 AV

A COOLING

7 HEATING, X/D INVERTED AV

AV

A 7

a

VA

B

VA

0

A V

1 1 1 1 i 1 ! 1 1 _ •0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

T

Figure 4.2. Comparison of the vertical temperature distribution within

the sphere during cooling with the inverted temperature

distribution during warming.

40
resistance should be expected because of the concomitant increase in the

wall temperature gradient.

The agreement between the heater and quasi-steady cooling temperature profiles is reasonably good for the hemispheres (figure CIO) and

poor for the vertical cylinder (figures C.13). Because the core of the

hemispheres is highly turbulent even in the quasi-steady cooling experiments, the additional turbulance added by the heater apparently has

little effect. In the case of the vertical cylinder, however, the heater

evidently causes considerable distortion of the natural flow patterns.

In both heater experiments, the maximum dimensionless temperature is

only slightly greater than in the cooling experiments.

4.2 Nusselt Numbers

The general agreement between the Nusselt numbers for the various

shapes (as indicated in figure 3.1) is not too surprising considering

T 28

1

that King obtained similar agreement for convection from vertical

cylinders and planes, horizontal wires, spheres, and blocks. In both

cases this agreement is probably due more to the comparison of surface

average heat transfer coefficients rather than to basic similarity in

[29] the details of the processes. For instance, Jakob has shown that the

average heat transfer coefficient from a horizontal plate is nearly the

same as for a vertical plate, in spite of the 2:1 variation in the

average heat transfer coefficient between the top and the bottom of the

horizontal plate. Because there is no theoretical justification for

using a single expression to represent the data for all the shapes, the

expressions given in the separate figures of Appendix C are generally

recommended

.

The bulk of the data is well represented by a Rayleigh number

exponent of 1/3, although in a few cases a somewhat lower value was

obtained. These lower exponents occur only for the data for a single

size vessel, and are probably the result of a combination of factors.

The strong temperature dependence of the physical properties is a possible cause, since higher heat fluxes occurred at higher temperatures;

use of some average boundary layer temperature would tend to increase

41
the exponent. Statistical variations in the slope can also occur, and

finally, some of the effect could be real. When both the size and the

heat flux are allowed to vary, however, an exponent near 1/3 best re- presents the process.

Comparing the stainless steel and copper-plastic hemisphere results

to the copper hemisphere results (table C.l), we note a general trend

toward a lower Nusselt number with increasing wall resistance. Because

the inside wall temperatures are calculated rather than measured, the

accuracy of these results is lower than for the copper-walled vessels,

and a correlation of Nusselt number vs. wall resistance is unfeasible.

The experiments using spatially distributed electric heaters show reasonable agreement of the Nusselt number with the quasi-steady cooling

results, although the slope of the Nu vs. Ra curves tend to be greater.

In comparing our Nusselt numbers with those in the literature, we

find that quasi-steady, turbulent, free convection within enclosed vessels

has been investigated neither for the cryogenic fluids studied here, nor

r 6

1

for other fluids. Schmidt, reporting on transient turbulent convection

8 11

(3 x 10 < Ra < 5 x 10 ) of water and alcohols within spheres, found

1/3 that Nu = 0.119 Ra using the average wall-to-bulk fluid temperature

difference. Evans and Stefany^ use the expression Nu = 0.55 Ra^ ^ to

5 9 fit their transient laminar (6 x 10 < Ra < 7 x 10 ) experiments of free

convection of water, glycerine, etc., within horizontal and vertical

cylinders. Both the Schmidt and the Evans and Stefany results are in

close agreem

King's data)

f 28 ] close agreement with the expressions recommended by Jakob (based on

1/4 Nu = 0.555 Ra for laminar flow,

and

1/3 Nu = 0.129 Ra '

for turbulent natural convection outside horizontal and vertical cylinders,

vertical planes, blocks, and spheres. In contrast, we obtain

42
1/3 Nu = 0.166 Ra when fitting

the and LN^ data with the exponent fixed at 1/3.

Because of the different fluids used and the different mode of heat

transfer (quasi-steady instead of transient) this positive 27 percent

deviation of our results from the classical Jakob expression is not

[9] entirely unexpected. In fact, Deaver and Eckert also observed a substantial positive deviation (28-percent at Ra = 10^) from the Jakob

expression for their quasi-steady laminar experiments with water and

3 7 ethylene glycol in horizontal cylinders (3 x 10 < Ra < 10 ) . Thus

enhanced heat transfer may be a characteristic of quasi-steady natural

convection.

In section 2.7 we note a 20 percent uncertainty in the thermal con- ductivity of LD^. In these experiments, however, the Nusselt numbers

for LD^ and LN_ are in excellent agreement, while the deviations of the

LH_ data from the other data are minus 5 percent for the 17.1-cm diameter

copper hemisphere, and minus 12 percent for the L/D = 2, vertical

cylinder. Since it is unlikely that the systematic error in the thermal

conductivity of all these fluids is in the same direction, and because

the properties of are better known than those of and , we

revise downward our estimate of the systematic error in the thermal

conductivity of ID^ from 20 percent to about 10 percent.

5. Summary and Conclusions

The goal of this study was to provide heat transfer design data for

liquid hydrogen and deuterium moderators and targets which are used in

high energy physics experiments. Because internal heat generation is

difficult to achieve in the laboratory, we simulate the internal heat

generation with a quasi-steady cooling process. Thus, quasi-steady

natural convection of U)^, LH^, and LN

2 within a sphere,, hemisphere,

horizontal cylinder, and vertical cylinder has been studied experimentally for the case of a nearly uniform wall temperature.

43
From this study we conclude:

1. A single expression,

0 352

Nu = 0.104 Ra

8 11 correlates the LN

2 and LH^ data over the range 7 x 10 < Ra < 6 x 10

with a standard deviation of 10 percent, whereas the hydrogen Nusselt

numbers are 8 percent lower. In all cases the thermal properties

listed in table 2.2 should be used in applying the correlation, since

it was developed using these property values. In applying this

correlation to other fluids, an additional systematic error of 10

percent could result because of the uncertainty in the thermal

properties (principally in the thermal conductivity) of liquid

deuterium.

2. An enhancement of heat transfer may be characteristic of quasisteady free convection within enclosures because the Nusselt numbers

[9] for both these results and those of Deaver and Eckert exceed the

Nusselt numbers typical of transient convection within enclosures

by about 27 percent.

3. The temperature field within the vessels is characterized by the

virtual absence of horizontal temperature gradients - except for a thin

boundary layer at the wall. For fully developed turbulent flow the

dimensionless vertical temperature profile is independent of Rayleigh

number for a given vessel shape. The temperature profiles for the

sphere, hemisphere, and horizontal cylinder are nearly identical

(figure 3.2), while the S profile for the vertical cylinders

is independent of L/D for L/D = 2 and 4.

4. The effect of wall resistance is small. Hemispheres having a significant wall resistance (greater wall temperature gradient)

exhibited only a small reduction in the Nusselt number compared to

the low wall resistance hemispheres.

5. Use of spatially distributed electric heaters gives Nusselt

numbers in reasonable agreement with the quasi-steady results,

44
although the slope of the Nu vs. Ra curves tends to be greater with

the heater. The agreement between the heater and the quasi-steady

cooling temperature profiles is good for the hemispheres, and poor

for the vertical cylinder. In both heater experiments the maximum

temperature is slightly overestimated.

Finally, correlation of the Nusselt numbers for the variety of

shapes tested here by a single expression, suggests that this correlation may apply to other shapes as well, provided they have a modest

aspect ratio. Further evidence in support of the possibly greater

generality of the results is the existence of a nearly universal

vertical temperature profile for vessels with both end and side wall

cooling (spheres, hemispheres and horizontal cylinders).

6. Acknowledgments

A number of people in the Cryogenics Division contributed to the

successful completion of this study. In particular I wish to thank:

R. 0. Voth and C. F. Sindt, who aided in the design of the apparatus;

L. M. Anderson, who built and helped operate the apparatus; and W. J. Hall

who wrote the data reduction programs.

45
7. Nomenclature

wall surface area , cm^

modified specific heat = C + B(i.-i)

P J proportionality constant in the Nusselt - Rayleigh number

correlation defined by Nu = Ra

n

proportionality constant defined by equations 7B and 12B.

specific heat at constant pressure

diameter

total energy

average heat transfer coefficient

specific enthalpy

thermal conductivity

cylinder length mass

Nusselt number = — = 3——

kA(T -T ) b w Rayleigh number exponent

pressure

heat transfer rate, watts

rate of internal heat generation per unit volume

heat transferred „ _ gPp C D (T

b

-Tw

) Rayleigh number = tL—^-

For vertical cylinders the characteristic length is L.

time

absolute temperature _ —

i . . -, w

dimensionless temperature =

T, - T

b w

specific internal energy

total internal energy

total volume

specific volume

46
w velocity

W work

W velocity vector

x position

Greek

1 ,dv, P thermal expansivity = —

V (g^)

P

9T*

9 dimensionless temperature gradient =

Q (x/L)

\i viscosity

p density

a standard deviation

Subscripts

b bulk fluid

c core w interior wall surface

47
8. References

1. H. Z. Barakat and J. A. Clark, Analytical and Experimental Study

of the Transient Laminar Natural Convection Flows in Partially Filled

Liquid Containers, Proc. 3rd Int. Heat Transfer Conf., Chicago 2^, 152-162 (1966).

2. J. D. Heliums and S. W. Churchill, Computation of Natural Convection

by Finite Difference Methods, Proc. 1961 Int. Heat Transfer Conf.,

985-994. ASME, New York (1961).

3. L. B. Evans, R. C. Reid, and E. M. Drake, Transient Natural Convection in a Vertical Cylinder, AIChE J., 14, 251-259 (1968).

4. M. J. Lighthill, Theoretical Considerations on Free Convection in

Tubes, Quart. J. Mech. Appl. Math, 6, 398-439 (1953).

5. F. G. Hammitt and Paul T. Chu, Transient Internal Natural Convection Heating and Cooling of Closed, Vertical, Cylindrical Vessels,

ASME No. 62-WA-309, 1-12 (1962).

6. E. Schmidt, Versuche zum Warmeubergang bei naturlicher Konvection,

Chem. Ing. Tech. 28, 175-180 (1956).

7. L. B. Evans and N. E. Stefany, An Experimental Study of Transient

Heat Transfer to Liquids in Cylindrical Enclosures, Chemical

Engineering Progress Symposium Series , 62 , No. 64, pp. 209-215.

AIChE, New York (1966).

8. H. G. Maahs, Transient Natural Convection Heat Transfer in a Horizontal Cylinder, Ph.D. dissertation, Chem. Library, U. of

Wash. (1964).

9. F. K. Deaver and E. R. G. Eckert, An Interf eromatic Investigation

of Convective Heat Transfer in a Horizontal Fluid Cylinder with

Wall Temperature Increasing at a Uniform Rate, Heat Transfer

1970 (Proc. 4th Int. Heat Transfer Conf., Paris-Versailles 1970),

Vol. IV, Elsevier, Amsterdam (1970).

10. J. A. Clark, A Review of Pressurization, Stratification, and

Interf acial Phenomena, International Advances in Cryogenic

Engineering , K. D. Timmerhaus (ed.), Sections M-U, pp. 259-283.

Plenum Press, New York (1965).

11. R. G. Schwind and G. C. Vliet, Observations and Interpretations of

Natural Convection and Stratification in Vessels, Proc . of the

1964 Heat Transfer and Fluid Mechanics Institute , W. H. Giedt

and S. Levy (eds.), pp. 51-68. Stanford University Press, Stanford,

Calif. (1964).

48
12. J. W. Tatom and W. 0. Carlson, Transient Turbulent Free Convection

in Closed Containers, Proc. 3rd Int. Heat Transfer Conf . , Chicago,

2, AIChE, N. Y. 163-171 (1966).

13. L. L. Sparks and R. L. Powell, Low Temperature Thermocouples:

kp, "normal" silver, and copper versus Au-0.02 at % Fe and Au

-0.07 at % Fe, J. Res. Nat; Bur. Stand. (U.S.), 76A (Phys . and

Chem.), No. 3, 263-283 (May-June 1972).

14. R. Prydz, The Thermodynamic Properties of Deuterium, Nat. Bur.

Stand. (U.S.) unpublished report (April 1967).

15. E. G. Brentari and R. V. Smith, Nucleate and Film Pool Boiling

Design Correlations for 0^, N , H_ and He, International Advances

in Cryogenic Engineering , K. D. Timmerhaus (ed.), Sections M-U,

pp. 259-283. Plenum Press, New York (1965).

16. J. P. Brouwer, A. M. Vossepoel, C. J. N. Van Den Meijdenberg, and

J. J. M. Beenakker, Specific Heat of the Liquid Mixtures of Neon

and Hydrogen Isotopes in the Phase Separation Region II. The

System Ne-D , Physica 50, pp. 125-148, North Holland Publishing

Company (19/0)

.

17. E. C. Kerr, E. B. Rifkin, H. L. Johnson, and J. T. Clarke, Condensed

Gas Calorimetry II. Heat Capacity of Ortho-deuterium Between

13.1 and 23.6 K Melting and Boiling Points, Heats of Fusion and

Vaporization. Vapor Pressure of Liquid Ortho-deuterium, J_^ Amer

.

Chem. Soc. 73 (1951)

.

18. V. G. Konareva and N. S. Rudenko, Measurements of the Viscosity of

Hydrogen Isotopes Along the Curve of Liquid-vapor Equilibrium,

Zh. Fiz. Khim . 41 No. 9 (1967).

19. H. M. Roder, G. E. Childs, R. D. McCarty, P. E. Angerhofer,

Survey of the Properties of the Hydrogen Isotopes Below Their

Critical Temperatures, Nat. Bur. Stand. (U.S.) Tech. Note 641

113 pages (Aug. 1973)

.

20. H. M. Roder, L. A. Weber, and R. D. Goodwin, Thermodynamic and

Related Properties of Parahydrogen from the Triple Point to

100°K at Pressures to 340 Atmospheres, Nat. Bur. Stand. (U.S.)

Monogr. 94, 110 pages (Aug. 1965).

21. R. D. McCarty and L. A. Weber, Thermophysical Properties of

Parahydrogen from the Freezing Liquid Line tc 5000 R for Pressures

to 10,000 psia, Nat. Bur. Stand. (U.S.) Tech. Note 617, 169 pages

(April 1972)

.

49
22. T. R. Strobridge, The Thermodynamic Properties of Nitrogen from

64 to 300°K Between 0.1 and 200 Atmospheres, Nat. Bur. Stand.

(U.S.), Tech. Note 129 (Jan. 1962).

23. H. J. M. Hanley, R. D. McCarty and W. M. Haynes, The Viscosity and

Thermal Conductivity Coefficients for Dense Gaseous Liquid Argon,

Krypton, Xenon, Nitrogen and Oxygen, J. Phys. Chem. Ref. Data _3 (9) (1974).

24. R. W. Powers, R. W. Mattox, and H. L. Johnston,' Thermal Conductivity

of Condensed Gases. II The Thermal Conductivities of Liquid

Normal and of Liquid Parahydrogen from 15 to 27 degrees K, J. Am.

Chem. Soc. Vol. 76, 5972-73 (1954).

25. H. M. Roder and D. E. Diller, Thermal Conductivity of Gaseous and

Liquid Hydrogen, J. Chem. Phys, 5_2, 11, 5928-49 (1970).

26. W. M. Rohsenow and H. Y. Choi, Heat, Mass, and Momentum Transfer,

p. 162, Prentice-Hall, Inc., Englewood Cliffs, New Jersey (1961).

27. Ibid., p. 204.

28. M. Jakob, Heat Transfer, Vol. 1, p. 530, John Wiley, New York (1959)

29. M. Jakob, Heat Transfer, Vol. 1, p. 532, John Wiley, New York (1959)

50
APPENDIX A

Heat Capacity During Cooling

As the vessel cools, fluid enters it at a temperature which is

generally cooler than the bulk temperature of the enclosed fluid. If we consider a control volume around the vessel, then the First Law of

Thermodynamics may be written as

dE = dQ - dW + I i.dm. (LA)

j

3 J

Noting that there is no external work done on the control volume,

dW = 0; (2A)

and equating E and U, we may write

dE = mdu + udm. (3A)

For a constant volume process, conservation of mass gives

dm = - S^L . (4A) v

Combining equations 1A through 4A, and dividing by dt to obtain the

time rate of change, we have

|| + -

v £ <v>i-- < 5A)

Noting that

du , du . , dT , dt "

( dr Wj '

and that

51
dv _ /dVx / dl\

dt " dT l dt j '

equation (5A) becomes

dT r du , 1 dv , . , , . N m dT [ dT +

v dT (l

i " U)] =

q - (6A)

For the constant pressure process assumed here

1 (—) = (3 v dT P

and

du

dT C

p

" = C

p

" P (1 " U)

Equation (6A) now becomes

m

of [C

p +

" 1)1 =

q; (?A)

or if we define a modified specific heat, C, as

c = c

p + - i),

we can finally write

q = mC g . (9A)

52
APPENDIX B

Velocity in the Core of a Vertical Cylinder

Consider a model (similar to that adopted by Evans, Reid and

[3] Drake ) , which consists of a slow moving cylindrical core with a uniform

vertical velocity, w , and a thin annular boundary layer. The vertical

velocity in the core is related to the vertical temperature gradient by

a common expression for both quasi-steady cooling and uniform internal

heat generation - as well as for the inverses of these processes. We

will consider the quasi-steady cooling core first.

B.l Quasi-Steady Cooling

For the control volume below, assuming no axial or transverse

conduction, an energy balance for a constant density fluid yields:

pwr C (T + ^— dx)

c p ox

I

I h — 1 9 T pdxC v ^— r— • control volume

p 9t 1

pw C T

c p

pwr C 1^ dx = pdxC

c p ox r p ot

w ^— = ^— (.lis; c ox ot

For quasi-steady state the temperature at all points within the vessel

changes uniformly, so that we may write

BT

P VC

P 5t - «

or,

Noting that

2

7TD AD

V = ^7- L = t—

4 4

53
we obtain

ai _ ^ • (2B) dt P C D

' (2B)

P

Combining equations (IB) and (2B) yields the core velocity, w^, as a function of the core temperature gradient and the wall heat flux:

4(q/A)

rPC D(f^)

p ox

W

c "— < 3B)

c

The core temperature gradient, (3T/9x) , may be eliminated from

c equation (3B) in the following manner. From the experiments, we note

that

3T*

a , /T , = Constant = 0 (4B) o (x/L)

over much of the length of the vertical cylinders. Since the dimensionless

temperature difference, T*, is defined as

T - T

T* = 1 , T, — T

b w

the core temperature gradient becomes

c

The characteristic temperature difference, T, - T , is eliminated

b w from equation (5B) using the heat transfer correlation

Nu = C^Ra

11

. (6B)

Employing the definitions of Nu and Ra, equation (6B) may be transformed

to

54
T

b

- r (q/A)

1 l-3n

n+1 „ n+1 w n+1 gPp

2

c

T n-1

|J.k n

n+1

or defining

C

2 " n+1

n-1

|ik n

n+1

(7B)

we obtain

1 l-3n

T

b

- Y

w

= C

2 (q/A)

n+1

D

n+1

(8B)

Combining equations (5B) and (8B) yields

1 l-3n

_ t, , . n+1 ~n+l

a

C (q/A) D ^ = —

ox iL

c

(9B)

Using (9B) to eliminate the core temperature gradient from equation (3B)

gives the core velocity as a function of the wall heat flux, the fluid

properties, and the geometry:

55
w

c

n 3n-l

4(L/D)(q/A) n+1

D

n+1 w

pc c„e

p 2

(10B)

For n = j, equation (10B) becomes

4(L/D)(q/A) 1/4 w

w pc c 0 e

P 2

(11B)

and is given by

C

2 =

(c

1

)

R 2,2_ ~] 1/4

3/4

(12B)

2 Evaluation of (11B) at a wall heat flux of 0.1 W/cm gives a core velocity of 0.5 cm/s for IJ^.

B.2 Internal Heat Generation

Following a fluid element as it rises in the core, and assuming

no axial or transverse conduction, an energy balance for a constant

density fluid gives

P C

3T

p 9t

(13B)

For steady state the core temperature gradient is

(91) = (91) ( 9t) = ( W W

9T) L_ K dx } K dt } w

c c

or rearranging,

9T ,dT,

^77 = w ) ot c ox

c

(14B)

56
Combining equations (13B) and (14B) yields

q w = . (15B)

PCr (^) p ox

c

For a cylinder insulated at the ends

4(q/A) w

and (15B) becomes

4(q/A) w

c = (16B)

pCr D(|^)

p ox

c

which is identical to the result for the quasi-steady cooling case,

equation (3B) . The remainder of the analysis is the same as in section

B.l.

57
APPENDIX C

Experimental Results

i i—i—

r

J I L

o<—

i

o

O

CO

4-1 C

cu I

u

CUa.

cubOG

•HH

oo

u

o

i

c

cu

CO

CO

CO

n

cuCuo

CJ

cuU

CU

Cu in

u

<uu

•H

COo IT)

CT>O CM I O

53
59
60
-0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

T*

Figure C.4. Horizontal cylinder (copper wall) - vertical temperature

distribution.

61
62
63
64
65
Figure C.9. Stainless steel and plastic-copper hemisphere - vertical

temperature distribution for cooling experiments.

66
T*

Figure C.10. Stainless steel and plastic-copper hemispheres with heaters -

vertical temperature distribution.

67
yi i i i i i r

I I i L

68
69
70

-0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

T*

Figure C.15. Horizontal cylinder (copper wall) - vertical temperature

distribution for warming experiments.

72
X/D

-0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

T*

Figure C.16. Copper hemisphere (D = 17.1 cm) - vertical temperature

distribution for warming experiments.

73
L/D

-0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

T*

Figure C.17. Vertical cylinders (copper wall) - vertical temperature

distribution for warming experiments.

74
Table C.l.

Correlation results using the expression Nu = C Ra

Test Vessel Fluid c,

1 n Std. Dev., %

in Nu

Sphere (copper) LD_

2

0. 446 0.284 4.2

D = 9.8 cm 0. 974 1/4 5.0

0. 140 1/3 5.9

Sphere (copper) LN_

2

0. 162 0. 327 7.0

D = 9.8 cm 0. 140 1/3 7.0

Copper Hemisphere LD

2

0. 470 0. 281 3.0 D = 6.5 cm 0. 935 1/4 3.6

0. 150 1/3 3.8

Copper Hemisphere LD

2

5. 19 0. 195 8.6

D = 17.1 cm 1. 32 1/4 9.1

0. 164 1/3 11.1

Copper Hemisphere LH2

0. 197 0.321 5.2 D = 17.1 cm 1. 16 1/4 6.0

0. 147 1/3 5 .

2

Stainless Steel Hemisphere LD2

0. 349 0. 297 10.2

D = 17. 1 cm 0. 141 1/3 10.4

Stainless Steel Hemisphere LD_

2

5. 15(10)" 5 0.645 6.3

With Heater D = 17.1 cm 0. 132 1/3 13.0

Plastic - Copper Hemisphere LD.

2

5. 03 0. 191 5.6 D = 17.1 cm 1, 17 1/4 6.3

0. 150 1/3 8.9

Plastic - Copper Hemisphere LD

2

0. 187 0.328 4.5

With Heater 0, 162 1/3 4. 5

Horizontal Cylinder (copper) LD,

2

0, 259 0.309 7.8

D = 7.4 cm 0.,151 1/3 8.0

Vertical Cylinder (copper) LD

2

0..109 0.352 10.8

D = 7.4 cm, L/D = 2 0., 171 1/3 10.8

Vertical Cylinder (copper) LH

2

0,,00554 0.379 4.0

D = 7.4cm', L/D = 2 0 .163 1/3 4.3

75
Table C . 1 Continued .

Correlation results using the expression Nu = C^Ra".

Test Vessel Fluid

J. n Std. Dev., % in Nu

Vertical Cylinder (copper) LN

2

0. 212 0.324 7. 2 D = 7.4 cm, L/D = 2 0. 170 1/3 7

,

2

Vertical Cylinder (copper) LD- 4. 51(10) 0.483 5. 0

T » 7 i 1 IT j. p. HA ~ . . T / T\ With neater, D - /.4 cm, L/

U

-

— 9L 1 . DC i / 1 1/ QO . LO

Vertical Cylinder (copper) LD

2

0. 0720 0. 366 5. 2

Pv — 7 A ^ »~ T / r> — A L) — 1 . 'i Cm , Li/ L) — 4 u

,

1 / U

1 / 7. J-/ J

rD . r5

Sphere (copper) LD

2

& LN

2

0. 376 0.291 5. 4 D = 9.8 cm 0, 140 1/3 6. 3

Copper Hemispheres LD. 0. 263 0.314 16. 6 D = 6. 5 cm, 17. 1 cm

1

0. 164 1/3 16,,9

Vertical Cylinders LD.

2

0. 152 0. 338 7 8 D = 7.4 cm, L/D = 2,4 0 171 1/3 7. 8

All Copper Vessels LD

2

0. 104 0.352 10. 7 Cooling 0. 166 1/3 11..0

All Copper Vessels All 0. 939 0. 356 10 .4 Cooling Fluids 0,, 165 1/3 10.,6

All Copper Vessels LD

2

0 . 104 0.355 17..1 Warming 0 , 177 1/3 17. 9

76
NBS-1 14A (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1 PlIBI [CATION OR RKPORT NO

NBS-IR 75-807

2• Oov t Accession

No.

3. Recipient's Accession No.

4.A TIT 1 1 1 1.1. I K AlMl^ A\in jUP 'slfRTITl MILLK TURBULENT NATURAL CONVECTION OF LIQUID

DEUTERIUM , HYDROGEN , AND NITROGEN WITHIN

ENCLOSED VESSELS

5. Publication Date February 1975

6. Performing Organization Code

7. AUTHOR(S)

David E. Daney

8. Performing Organ. Report No.

DUW LTHD M1MT ODT AMf7 ATIHM M A fc/T7 A MPl A nPiD I7CC T. rtKrUKMHNU UKOa1\IZ,A 1 iUIN NAMl AINU ADUKhoc*

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 2754450

11. Contract/Grant No. SANL 351

\ L . Snnnsnnnp ( )rp;in i 7 flf i nn N 3 m f a n r i ( nmnlprp Aiirlr^»cc f^tre^e^t (~*

1 «.

1 i i\r ^ t a t o 71 P \ vj 1 1 r> i ill

f-, V'l ^ an i^. ai i uil 11 aim "11 U ^— ClllUltriCT "UUI CaS l>Jll Ctrl, V—' I ly , •^talC, / 1 J Lawrence Livermore Laboratory

University of California

P. O. Box 808

Livermore T California 94550

13. Type of Report & Period Covered

Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less (actual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

Quasi-steady natural convection of liquid deuterium, hydrogen, and

nitrogen within a sphere, hemisphere, horizontal cylinder, and vertical

cylinder has been studied experimentally for the case of a nearly

isothermal wall. A single expression relating the Nusselt and Rayleigh

numbers,

Nu = 0.104 Ra , 8 11 fits the deuterium and nitrogen data over the range 7 x 10 < Ra < 6 x 10 , while the hydrogen Nusselt numbers are 8 percent lower. The temperature

field within the vessels is virtually free of horizontal temperature

gradients. A single dimensionless temperature profile characterizes the

vertical temperature distribution for each vessel shape, with the profiles for the sphere, hemisphere, and horizontal cylinder being nearly

identical.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons) Cryogenic; heat transfer; hemisphere; horizontal cylinder;

liquid deuterium, liquid hydrogen, liquid nitrogen; natural convection;

sphere; vertical cylinder.

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