cryogenics,THERMAL PROPERTIES

2.6. Thermal conductivity

The thermal conductivity k_t of a material is defined as the heat-transfer rate per unit area divided by the temperature gradient causing the heat transfer. The variation of thermal conductivity of several solids is shown in Fig. 2.7. Values of the thermal conductivity of cryogenic liquids and gases are given in Appendixes B through E.

Fig. 2.7. Thermal conductivity of materials at low temperatures: (ll 2024-T4 aluminum: (3) beryllium copper; (3) K Monel; (4) titanium; (5) 304 stainless steel; (6) C1020 carbon steel; (7) pure copper; (8) Teflon (Stewart and Johnson 1961).

To understand the variation of thermal conductivity at low tempera­tures, one must be aware of the different mechanisms for transport of energy through materials. There are three basic mechanisms responsible for conduction of heat through materials: (l) electron motion, as in metallic conductors; (2) lattice vibrational energy transport, or phonon motion, as in all solids; and (3) molecular motion, as in organic solids and gases. In liquids, the primary mechanism for conduction heat trans­fer is the transfer of molecular vibrational energy; whereas in gases, heat is conducted primarily by transfer of translational energy (for monatomic gases) and translational and rotational energy (for diatomic gases).

For all gases the thermal conductivity decreases as the temperature is lowered. Because the product of density and mean free path for a gas is practically constant, and the specific heat is not a strong function of tem­perature, the thermal conductivity of a gas should vary with temperature in the same manner as the mean molecular speed u, as indicated by eqn. (2.3). From kinetic theory of gases (Present 1958), the mean molecular speed is given by

All cryogenic liquids except hydrogen and helium have thermal con­ductivities that increase as the temperature is decreased. Liquid hydrogen and helium behave in a manner opposite to that of other liquids in the cryogenic temperature range.

For heat conduction in solids, the thermal conductivity is related to other properties by an expression similar to that for gases,

Energy is transported in metals by both electronic motion and phonon motion; however, in most pure electric conductors, the electronic contri­bution to energy transport is by far the larger for temperatures above liq­uid-nitrogen temperatures. The electronic specific heat is directly propor­tional to absolute temperature (see Sec. 2.7), and the electron mean free path in this temperature range is inversely proportional to absolute tem­perature. Because the density and mean electron speed are only weak functions of temperature, the thermal conductivity of electric conductors above liquid-nitrogen temperatures is almost constant with temperature, as would be predicted by eqn. (2.5). As the absolute temperature is low­ered below liquid-nitrogen temperatures, the phonon contribution to the energy transport becomes significant. In this temperature range, the ther­mal conductivity becomes approximately proportional to T-² for pure metals. The thermal conductivity increases to a very high maximum as the temperature is lowered, until the mean free path of the energy carriers becomes on the order of the dimensions of the material sample. When this condition is reached, the boundary of the material begins to intro­duce a resistance to the motion of the carriers, and the carrier mean free path becomes constant (approximately equal to the material thickness). Because the specific heat decreases to zero as the absolute temperature approaches zero, from eqn. (2.5) we see that the thermal conductivity would also decrease with a decrease in temperature in this very low tem­perature region.

In disordered alloys and impure metals, the electronic contribution and tire. phonon contribution to energy transport are of the same order of magnitude. There is an additional scattering of the energy carriers due to the presence of impurity atoms in impure metals. This scattering effect is directly proportional to absolute temperature. Dislocations in the mate­rial provide a scattering that is proportional to T², and grain boundaries introduce a scattering that is proportional to T³ at temperatures much lower than the Debye temperature. All these effects combine to cause the thermal conductivity of allays and impure metals to decrease as the tem­perature is decreased, and the high maximum in thermal conductivity is eliminated in alloys.

2.7. Specific heats of solids

The specific heat of a substance is defined as the energy required to change the temperature of the substance by one degree while the pressure is held constant (c_r) or while the volume is held constant (c_v). For solids and liquids at ordinary pressures, the difference between the two specific heats is small, while there is considerable difference for gases. The vari­ation of the specific heats with temperature gives an indication of the way in which energy is being distributed among the various modes of energy of the substance on a microscopic level.

Specific heat is a physical property that can be predicted fairly accu­rately by mathematical models through statistical mechanics and quan­tum theory. For solids the Debye model gives a satisfactory representa­tion of the variation of the specific heat with temperature. In this model, Debye assumed that the solid could be treated as a continuous medium, except that the number of vibrational waves representing internal energy must be limited to the total number of vibrational degrees of freedom of the atoms making up the medium-that is, three times the total number of atoms. The expression for the specific heat of a monatomic crystalline solid as obtained through the Debye theory is

      where 8_D is called the Debye characteristic temperature and is a property of the material, and D(T/q_D) is called the Debye function. A plot of the specific heat as given by eqn. (2.6) is shown in Fig. 2.8, and the Debye specific-heat function is tabulated in Table 2.1. Values of q_D for several substances are given in Table 2.2.

Table 2.1. Debye specific heat function

T/qD	Cv/R	T/qD	Cv/R	T/qD	Cv/R
0.08 0.09 0.10 0.12 0.14 0.16 0.18 0.20 0.25 0.30 0.35 0.40	0.1191 0.1682 0.2275 0.3733 0.5464 0.7334 0.9228 1.1059 1.5092 1.8231 2.0597 2.2376	0.45 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50	2.3725 2.4762 2.6214 2.7149 2.7781 2.8227 2.8552 2.8796 2.8984 2.9131 2.9248 2.9344	1.60 1.70 1.80 1.90 2.00 2.20 2.40 2.60 2.80 3.00 4.00 5.00	2.9422 2.9487 2.9542 2.9589 2.9628 2.9692 2.9741 2.9779 2.9810 2.9834 2.9844 2.9900

2.8 Specific Heat of Liquids and Gases

In general, the specific heat C_v of cryogenic liquids decreases in the same way that the specific heat of crystalline solids decreases as the tempera­ture is lowered. At low pressures the specific heat c_p decreases with a decrease in temperature also. At high pressures in the neighborhood of the critical point, humps in the specific-heat curve are observed for all cryogenic fluids (and in fact for all fluids).

                                                                                (2.10)

In the vicinity of the critical point, the coefficient of thermal expansion

becomes quite large; therefore, one would expect large increases of the specific heat c_p in the vicinity of the critical point also.

The specific heat of liquid helium behaves in a peculiar way-it shows a high, sharp peak in the neighborhood of 2.17 K (3.91^oR). The behavior of liquid helium is so different from that of other liquids that we shall devote a separate section to a discussion of its properties.

Gases at pressures low compared with their critical pressure approach the ideal-gas state, for which the specific heat C_u is independent of pres­sure. According to the classical equipartition theorem, the specific heat of a material is given by

C_v = ½Rf                                                                    (2.11)

where f is the number of degrees of freedom of a molecule making up the material. For monatomic gases, the only significant mode of energy is translational kinetic energy of the molecules, which involves three degrees of freedom. From eqn. (2.11) the specific heat C_v for such mona­tomic gases as neon and argon in the ideal-gas state is C_v = ³/₂R. In diatomic gases, other modes are possible. For example, if we consider a "rigid-dumbbell" model of a diatomic molecule such as nitrogen (i.e., we neglect vibration of the molecule), there are three translational degrees of freedom plus two rotational degrees of freedom. We consider only two rotational degrees of freedom because the moment of inertia of the molecule about an axis through the centers of the two atoms making up the molecule is negligibly small compared with the momel1t of inertia about an axis perpendicular to the interatomic axis. From eqn. (2.11) the spe­cific heat C_v for the rigid-dumbbell model of an ideal diatomic gas would be C_v = ⁵/₂R. This is true for most diatomic gases at ambient temperatures, at which the gases obey classical statistics. Because the molecules in a diatomic gas are not truly rigid dumbbells, we should also expect to have vibrational modes present, in which the two molecules vibrate around an equilibrium position within the molecule under the influence of the inter­atomic forces holding the molecule together. In this case, two more degrees of freedom due to the vibrational mode would be present, so the specific heat for a vibrating-dumbbell molecule would be C_v = ~R accord­ing to the classical theory.

In the actual case, the rotational and vibrational modes are quantized, so they are not excited if the temperature is low enough. The variation of the specific heat of diatomic hydrogen gas with temperature is illustrated in Fig. 2.9. At very low temperatures, only the translational modes are excited, so the specific heat C_v takes on the value ³/₂R, the same as that of a monatomic gas. To determine whether the rotational modes will be excited, one must compare the temperature of the gas with a character­istic rotation temperature q_r,

                                                                             If the temperature is less than about ¹/₃q_r the rotational mode is not appre­ciably excited; if the temperature is greater than about 3q_r, the rotational mode is practically completely excited. Most diatomic gases become liq­uids at temperatures higher than 3q_r; however, H₂, D₂, and HD are excep­tions to this statement. Because of the small moment of inertia of the hydrogen molecule, the characteristic rotation temperature is quite a bit above the liquefaction temperature for hydrogen (q_r = 85.4 K or 153.7^oR for hydrogen, so 3q, = 256.2 K or 461.1^oR). The specific heat C_v of hydro­gen gas rises from ³/₂R at temperatures below about 30 K (54°R) to ⁵/₂R at temperatures above 255 K (459°R).

The vibrational modes for a diatomic gas are also quantized and become excited at temperatures on the order of a characteristic vibration temperature q_v defined by

                   Fig. 2.9. Variation of the specific heat c, for hydrogen gas.

where f_v is the vibrational frequency of the molecule. The characteristic vibration temperature for gases is much higher than cryogenic tempera­tures, so the vibrational mode is not excited for gases at cryogenic tem­peratures (q_v = 6100 K or 1O,980^oR for hydrogen gas).

The change in the specific heat of hydrogen between 30 K and 255 K is important for hydrogen liquefiers and hydrogen-cooled helium lique­fiers because it affects the effectiveness of the heat exchangers, as we shall see in Chap. 3.

At pressures higher than near ambient, the specific heats of gases vary in a more complicated manner with temperature and pressure. A com­plete coverage of this effect is beyond the scope of our present discussion.

2.9. Coefficient of thermal expansion

The volumetric coefficient of thermal expansion b is defined as the frac­tional change in volume per unit change in temperature while the pres­sure on the material remains constant. The linear coefficient of thermal expansion b_t is defined as the fractional change in length (or any linear dimension) per unit change in temperature while the stress on the mate­rial remains constant. For isotropic materials, b = 3 l_t. The temperature variation of the linear coefficient of thermal expansion for several mate­rials is shown in Fig. 2.10.

The temperature variation of the coefficient of thermal expansion may be explained through a consideration of the intermolecular forces of a material. The intermolecular potential-energy curve, as shown in Fig. 2.11, is not symmetrical. Therefore, as the molecule acquires more energy (or as its temperature is increased), its mean position relative to its neigh­bors becomes larger; that is, the material expands. The rate at which the mean spacing of the atoms increases with temperature increases as the energy or temperature of the material increases; thus, the coefficient of thermal expansion increases as temperature is increased

Fig. 2.10. Linear coefficient of thermal expansion for several materials at low temperature: (I) 2024-T4 aluminum; (2) beryllium copper; (3) K Monel; (4) titanium; (5) 304 stainless steel; (6) C1020 carbon steel (NBS Monograph 29, Thermal Expansion of Solids at Low Temperatures).

Since both the specific heat and the coefficient of thermal expansion are associated with intermolecular energy, one might expect to find a rela­tionship between the two properties. For crystalline solids,

. Values of the Gruneisen constant for some materials are presented in Table 2.4.

Table 2.4. Values of the Gruneisen constant for selected solids

Material	gG
Aluminum	2.17
Copper	1.96
Gold	2.40
Iron	1.60
Lead	2.73
Nickel	1.88
Platinum	2.54
Silver	2.40
Tantalum	1.75
Tungsten	1.62

We have seen previously that the bulk modulus B is not strongly dependent upon temperature for solids (it increases somewhat as the tem­perature is decreased); therefore, the coefficient of thermal expansion for solids should vary with temperature in the same way that the Debye spe­cific heat varies with temperature. This general variation has been found true experimentally. At very low temperatures (T < q_D/12), the coeffi­cient of thermal expansion is proportional to T³•

References

1. Cryogenic Systems               -             Barron R. F
2. Cryogenic Engineering           -           Scot R. W.
3. Cryogenic Engineering           -           Bell J.H.