We consider means for using the cooling available in boil-off gas to intercept heat conducted through the support structure of a cryogen tank. A one-dimensional model of the structure coupled to a gas stream gives an analytical expression for heat leak in terms of flow rate for temperature independent-properties and laminar flow. A numerical model has been developed for heat transfer on a thin cylindrical tube with an attached vent line. The model is used to determine the vent path layout that will minimize heat flow into the cryogen tank. The results are useful for a number of applications, but the one of interest in this study is the minimization of the boil-off in large cryopropellant tanks in low Earth and low lunar orbit

Canavan, Edgar R.

Miller, F. K.

NASA Technical Reports Server

OPTIMIZED HEAT INTERCEPTION FOR CRYOGEN 
TANK SUPPORT 
E. R. Canavan and F. K. Miller 
Code 552 - Cryogenics and Fluids Branch 
NASA - Goddard Space Flight Center 
Greenbelt, Maryland, 20771, USA 
ABSTRACT 
We consider means for using the cooling available in boil-off gas to intercept heat 
conducted through the support structure of a cryogen tank. A one-dimensional model 
of the structure coupled to a gas stream gives an analytical expression for heat leak in 
terms of flow rate for temperature independent-properties and laminar flow. A 
numerical model has been developed for heat transfer on a thin cylindrical tube with 
an attached vent line. The model is used to determine the vent path layout that will 
minimize heat flow into the cryogen tank. The results are useful for a number of 
applications, but the one of interest in this study is the minimization of the boil-off in 
large cryopropellant tanks in low Earth and low lunar orbit. 
KEYWORDS: cryogenics; tank support; heat interception 
INTRODUCTION 
Optimizing heat interception in cryogen storage systems is an old problem in 
cryogenic engineering, and there have been many important advances in this area. Bejan 
and Smith [I] applied entropy generation minimization to the problem of cooling a 
structural support with gas streams. Bejan [2] extended this result to the case of discrete 
cooling points along the support. Chato and Khodadi [3] considered a system with both 
structural supports and cooled shields. More recently, Hofmann [4] derived a solution for 
the number and placement of shields cooled by boil-off gas. 
These optimizations however, are generally applicable to ground-based cryostats. 
Cryogens tanks for space have the additional requirement that the overall system mass be 
minimized. Thus, the mass of any additional hardware for reducing the heat leak, such as 
cooled shields, must be weighed against the mass of simply making tanks large enough to 
achieve the required lifetime with a higher boil-off. Shields, must withstand transmitted 
vibration and acoustic loads, and must meet minimum mode frequency requirements. This 
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' causes the thickness, or mass per unit area, to increase with scale. Thus, shields become 
b less attractive at large scales. They are also less beneficial for fluids with higher heat of 
vaporization. However, use of boil-off gas to cool tank structural supports has little mass 
penalty and appears to be attractive at any scale. 
Here we consider the design of a practical heat exchanger for cooling the support 
structure, chiefly for large tanks. High heat transfer between the gas and the support is 
implicit in other work on optimized cooled supports. However, in a spaceflight tank, 
especially cryopropellant tanks that have no vacuum shell, it is likely that the vent tube will 
need to be sized to carry the initial, high heat load that the tank will experience as it reaches 
orbit. Thus, as the tank reaches steady state, the flow velocity in the tube will likely be 
well below the critical Reynolds number, so the dependence on the heat transfer coefficient 
needs to be examined. 
ONE DIMENSIONAL ANALYSIS 
The one-dimensional case provides valuable insight into the problem, because under 
the assumption of temperature-independent property values, it yields an analytic solution. 
Figure 1 shows a schematic of the geometry. A tube carrying cold gas flows parallel to the 
support. If the thickness of the support is small or if its conductance is high in the direction 
normal to the flow, it will be isothermal in planes normal to the flow, and the temperature 
of the support Ts, will be a function only of the distance along the flow direction, x. 
Assuming constant thermal conductivity, ks, and cross sectional area, A, the energy balance 
for the support is 
where h is the heat transfer coefficient, P is the tube perimeter, and Tf is the fluid 
FIGURE 1. Heat balance for a one-dimensional support with continuous heat exchange to flowing gas. 
temperature. This equation describes the simple one-dimensional fin problem in most heat 
transfer textbooks, except that the fluid temperature is here also a function of position. 
Assuming constant heat capacity, the enthalpy balance for the fluid yields 
where m is the mass flow rate of the fluid. We assume fixed boundary temperatures on the 
support, and a fluid temperature at the entrance fixed to the support boundary temperature: 
T,(x = 0)  = Tow, 
T, (x = L)  = Thigh, and 
T~ (X = 0 )  = Tow. 
Since the problem is linear, we can set Tl0, = 0, and normalize temperatures by Thigh: 
z = TIThigh. Replacing x with the non-dimensional = xlL yields 
hPL 
where /3 = - hPL is the number of transfer units, and a2 = -. The solutions can be put 
mCp kAIL 
in a reasonably compact form using the parameter = 4 2  + @: 
The heat leak into the cold end is 
Note that the factor k(A/L)Thigh is the heat leak down the support with no fluid flow. Thus, 
the non-dimensional heat leak, 
+ gives the ratio of the heat with and without fluid flow, and is a function only of the 
parameters y and p, or a and p. It is instructive to put qleak in terms of a new variable, 
p kAIL y = = -  , the ratio of the conductance to the capacity rate. 
a' mCp 
Figure 2 shows a contour plot of the non-dimensional heat leak as a function of /3 and 
y. White represents a region where qleak is close to 1; that is, a region where the heat 
conducted down the support is affected very little by the gas flow. Black represents a 
region where qleak is close to 0; that is, a region where the heat conducted down the support 
is nearly completely suppressed by the gas flow. It is clear that when the number of 
transfer units, p > 10, qleak depends only on y. In this case, when the support conductance 
is much greater than the capacity rate, y >> 1, the effect of the fluid is negligible, and when 
the conductance is much less than the capacity flow rate, y << 1, the conductance of the 
tube is largely suppressed. 
How does the behavior of /? and y scale with tank size? If the radius, R, and length of 
a cylindrical support scale by the same factor, kAIL depends only on thickness, t. Critical 
buckling load determines the thickness. Using the classical equation for the critical 
buckling load of a thin cylinder and setting it equal to the axial stress due to acceleration of 
the tank yields 
where m is the mass of the tank, x is the maximum acceleration, E is the Young's modulus 
of the support tube, and v is the Poisson ratio. Since mass scales as R3, t scales as R3". 
The capacity rate, mCp, is proportional to the heat load on the tank, which under most 
circumstances is dominated by radiation, which is in turn proportional to surface area and 
FIGURE 2. Suppression of the conductive heat leak in the one-dimensional case as a function of number of 
transfer units, p, and the ratio of conductance to capacity flow, p. When P >> 10, the suppression is only a 
function of p. 
scales as R2. Thus, ,u scales as R"", and larger tanks will have lower ,u. For P, the product 
hP is independent of scale for laminar flow. The length L scales as R, so P scales as R-'. 
SUPPORT GEOMETRY 
Three types of insulating support structures are generally used for cold space flight 
hardware: straps, hexapod struts, and cylindrical tubes. While straps have the smallest 
cross-sectional area, they put the warm structure in compression, and are thus not mass 
efficient for large tanks. The heat conduction in a hexapod strut is determined by the 
requirement that the resonant frequency of the system must exceed some minimum value. 
For the case of a spherical tank of radius R with struts attached at the radius of the tank and 
at the same radius on the base plate, the area-to-length ratio of each strut is given by 
where m is the mass of the tank, w = 2nf is the minimum permitted resonance frequency of 
the horizontal mode, E is the Young's modulus, and 5 = L/R, where L is the length from the 
center of the base plate to the center of the tank. A similar equation can be derived for a 
long, slender cylinder (using beam equations), but for the more practical case of short, 
squat cylinders, shear dominates over bending and A/L is simply given by mco2/~. 
Approximating the shear modulus, G = ED(l+v), the ratio of the conductances is 
A I L I  
strut = 4(1+ c2) 
AIL1 CY 1 (2 + 5 t 2  - 44 - l o t2  + 25c4)(l + v) ' 
where Y is Poisson's ratio. This function has a minimum at 2.3 (for Y = 0.2), so a strut 
support is always more than twice as conductive as a tube support with the same resonant 
frequency. Unfortunately, the tube thickness derived from the resonant frequency criterion 
is generally much too thin to withstand the local compressive stresses. Instead, thickness, t ,  
is determined by the requirement that stress must remain below the critical load for local 
buckling, as given in equation (8). Thus, A/L = 2nt/5 scales as ml", and there is some tank 
mass, 
below which struts give lower conductance, and above which a cylinder is superior. This 
analysis does not include the nonlinear correlation factor that takes into account 
degradation of buckling strength relative to the ideal limit due to imperfections [ 5 ] ,  and it 
does not consider lateral loads, for which buckling strength is somewhat lower. However, it 
does only consider thin cylindrical shells; higher buckling limits can be achieved with 
sandwich construction or stiffeners. 
FIGURE 3. Cylindrical support cooled by helical vent tube attached continuously along its length. 
CYLINDER WITH HELICAL COOLING TUBE 
For a thin squat cylindrical support, the obvious geometry for intercepting heat leak by 
a stream of gas is with a helical tube along its surface as shown in Figure 3. Conduction in 
a thin cylinder can be solved by treating it as a rectangular surface with a periodic 
boundary condition along the edges parallel to the axis. Unfortunately, the boundary 
conditions for such solutions must be fixed, and cannot be a function of another variable, as 
in this case, where the fluid in the helical tube is described by another differential equation. 
Therefore, we must resort to numerical techniques. We use a standard finite difference 
scheme. The cylinder is broken into an array of Nx elements in the circumferential 
direction and NxNwraps in the axial direction, where Nwraps is the number of complete turns 
the helical flow path makes around the cylinder. (We only treat the case where Nwraps is an 
integer.) Thus, there are N,, square arrays of elements and N,' N,,, equations for heat 
flow in the cylinder. The convective terms (equivalent to the right hand side of equation 
(1)) appear in the diagonal elements of each of these square arrays. The enthalpy balance 
equations for the fluid (equation (2)) link these NXNwraPs terms. Under the assumption of 
constant property values, all equations are linear, so a solution can be obtained by inverting 
the (N,N,raps(Nx + 1)) x (NxNwraps(NX + 1)) coefficient matrix. We use the same boundary 
conditions as for the one-dimensional case: fixed temperatures along the hot and cold edges 
of the cylinder, and the inlet temperature of the gas fixed at the cold boundary temperature. 
For temperature dependent property values, iterating the solution works quite well, with 
excellent convergence being achieved in roughly 20 iterations, and no relaxation being 
necessary. Tests show that as NX increases above 50, there is little change in the solution. 
The algorithm was implemented with MathematicaTM on a Macintosh Powerbook G4. A 
nonlinear (temperature-dependent property) solution typically took less than a minute. 
(Note: It was essential to use SparseArray representation, which exploits UMFPACK 
multifrontal direct solver methods, to achieve this performance.) 
When property values are taken as independent of temperature, the non-dimensional 
heat leak through the cylinder, qleak, is a function of the same two variables as in the one- 
dimensional case, ,u and P, plus two additional geometric variables, the number of wraps, 
Nwraps, and the cylinders aspect ratio, c. Typically, one would not use support tubes with a 
very large or small aspect ratio. The examples given below all use an aspect ratio 1.6. For 
FIGURE 4. Comparison of the 0.5 contour of heat leak suppression, q,,,k, determined by the inear 
(temperature-independent material property value) solution for Nwra,, = 1,2,  ..., 10, as well as the one- 
dimensional solution. a) Curves shown on a plot with the same range as Figure 2, for comparison. b) The 
same curves plotted on a narrower range of p. Note that as Nwrap, increases, it rapidly converges to a curve 
that is close, but not equal, to that of the one-dimensional solution. 
a given value of Nwraps, we can plot a contour map of qleak versus ,u and P, as is shown for 
the one-dimensional solution in Figure 2. These maps look quite similar. 
To better see the differences, and to understand the effects of increasing the number of 
wraps of the cooling loop, we plot only the qleak = 0.5 contour for Nwraps = 1, 2.. . ,10 in 
Figure 4, along with the qleak = 0.5 contour from the one-dimensional solution. The figure 
shows a significant improvement in performance in going from one to two wraps. That is, 
for the same thermal conductance to capacity flow rate, two wraps produces a greater 
suppression of the heat leak than one. However, beyond two wraps, the contours quickly 
converge. Interestingly, it converges to a line close, but not equal to, the one-dimensional 
case. In the region p >> 10, where qleak becomes independent of P, qleak = 0.5 occurs at p = 
0.76 for the cylinder, but ,u = 0.80 for the one-dimensional solution. In general, the one- 
dimensional solution is a very good approximation to the two-dimensional numerical 
solution for Nwraps 2 2. Note that numerical solution treats the number of transfer units, 
hPL p=- , and Nwraps as independent variables, but for a constant pipe cross section, hPL 
mC, 
will increase with the number of wraps. However, hP can often be made large with clever 
internal geometry. As noted above, if P >> 10, the solution is nearly independent of P. 
The temperature-dependent properties solution is not linear in temperature, so it is not 
a function of a few simple non-dimensional variables. A typical solution is shown in 
Figure 5. For this case, the support tube was GI0 with a radius of 406 rnm, a length of 640 
mm, and a thickness of 1 rnm. The number of wraps is 2, and the boundary conditions 
were Thigh = 100 K and Tlo, = 24.6 K. The cooling gas was hydrogen flowing at a rate of 
0.69 mgls. This flow rate was sufficient to strongly affect the temperature profile: the 
thermal conductivity of G10, as that of most materials, increases with increasing 
temperature, producing a temperature profile that is flat at the high temperature end, and 
decreases more sharply at the low end. The flow reverses this curvature, producing a 
profile that has a lower gradient, and thus lower heat flow, at the cold end. The calculated 
heat leak into the cold end is 13.8 mW, which is 13.5% of the heat leak with no gas flow. 
2.6 Circumferential 
\ position, m 
0.6 0.3 0 
axial position, m 
FIGURE 5. Temperature as a function of position on the cylinder. Top surface - nonlinear (temperature 
dependent material property value) solution. Bottom surface - linear (temperature independent material 
property value) solution. The linear solution is everywhere lower than the nonlinear, except at the upper and 
lower edges. 
Also shown in the figure is the temperature-independent properties solution. In this 
example, the temperature independent solution is everywhere lower, and it generates a 
significantly lower heat value, 10.9 mW, or 10.6% of the heat leak with no flow. (The 
average thermal conductivity is chosen so that the heat leak with no flow is the same in the 
linear and non-linear solutions.) There are some situations where the linear (temperature- 
independent properties) solution gives a higher heat leak value than the nonlinear, although 
the computed heat leaks are close to those computed from the full nonlinear solution. 
CONCLUSIONS 
For supporting large cryogen tanks against severe launch loads, a cylindrical support 
is superior to a hexapod strut above a certain size. Reducing the heat flow into the tank can 
be accomplished by attaching the vent line in a helix along the surface of the cylinder, so 
that the boil-off gas intercepts the inflowing heat. Modeling of this geometry shows that a 
helix of two wraps effectively intercepts the heat, and that with two or more wraps, the 
results match well to a simple one-dimensional analytic solution. Including the 
temperature dependence of the material property values has a significant, but not 
overwhelming, effect on the computed heat leak. 
REFERENCE 
1. Bejan, A. and Smith, J. L., Jr., Cryogenics 14, pp. 158-1 63 (1974). 
2. Bejan, A., Cryogenics 15 pp.290-292 (1975). 
3.  Chato, J. C. and Khodadi, J. M., J. Heat Transfer 106, pp. 871-875 (1984). 
4. Hofmann, A., Cryogenics, 44, pp. 159-165 (2004). 
5 Weingarten, V.I., et al., NASA SP-8007: Buckling of Thin- Walled Circular Cylinders, NASA (1968). 


Optimized Heat Interception for Cryogen Tank Support

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