Heat And High Temperature

Heat conduction. The steady state. The rate (expressed in calories per second) at which heat flows across an isothermal surface element of area A, in a homogeneous medium, is proportional to A, to the conductivity of the material, K, and to the temperature gradient dT/dx perpendicular to the surface, thus:

calories/sec. (1)

In the case of a rectangular parallelopiped with opposite ends maintained at the temperatures T₁ and T₂, Eq. 1, when integrated, becomes

calories/sec., (2)

in which A is the cross-section area of the parallelopiped perpendicular to the temperature gradient and x is the separation between the isothermal surfaces T₁ and T₂. Here K is assumed to be constant in the temperature range between T₁ and T₂. Values of the heat conductivity for various materials are given in Table I.

Shape factors. For many of the actual cases encountered, the geometry is not so simple as it is with the parallelopiped, and the integration of Eq. 1 may be quite difficult. Generally, however, the integral may be expressed by an equation of the form

calories/sec., (3)

in which S, the so-called shape factor, depends upon the size and shape of the space between two isothermal surfaces is maintained at temperatures T₁ and T₂.

For a rectangular parallelopiped the shape factor, from Eq. 2, is

(4)

For two concentric cylindrical isothermal surfaces (long in comparison with their radii) of length l, maintained at temperatures T₁ and T₂, respectively, the value of S in terms of l and their radii r₁ and r₂ is

. (5)

For two concentric spherical isothermal surfaces, one of radius r₁ at temperature T₁ and the other of radius r₂ at temperature T₂, the shape factor is

. (6)

A heat problem which often arises in the laboratory requires the estimation of the heat loss of an electric furnace. The inner furnace wall, which is approximately at uniform temperature, is taken as one isothermal surface, and the outer surface of the furnace, at somewhat above room temperature, is taken as the other isothermal surface. Langmnir, Adams, and Meikle have given shape factors for several special cases which may be applied to problems. Of this type.¹ However, to make an estimate of heat loss of a cylindrical furnace, if the inner furnace tube is long and surrounded by a layer of insulating material as shown in Fig. 1, we may apply Eq. 5. Or, for the case shown in Fig. 2, we may apply Eq. 6, taking r₁ and r₂ as the dimensions of the approximating spherical surfaces, indicated in the figure by dotted lines. These estimates are not expected to be precise, but they are usually accurate enough to settle the questions which arise when one designs a furnace.

The shape factor can also be determined experimentally, using the similarity between the law for the flow of heat, Eq. 1, and Ohm's law. The experimental determination of S is accomplished by measuring the electrical shape factor, S', for wooden models that simulate the inner and outer isothermal temperature surfaces of the heat problem in question. These model surfaces are coated with copper foil and serve as electrodes. The region between these surfaces is filled with a saturated solution of copper sulphate with per cent (by volume) sulphuric acid added. The conductance of this solution is determined by applying alternating voltage to the copper electrodes. Alternating current is used to prevent polarization at the electrodes. The equation giving the electrical shape factor is

, (7)

V and i being the measured voltage drop and current. K', the electrical conductivity of the solution, may be determined by a separate experiment, using a box of cross section A' and length x' with copper end plates. For this box the shape factor is A'/x' (as in Eq. 4). To transform S' to S, divide S' by the scale factor to which the model was constructed. For example, if the model was made to half scale, S = 2S'.

Heat conduction. The nonsteady state. The thermal behavior of a homogeneous body is described in a Cartesian system of coordinates by the following fundamental differential equation:

. (8)

Here t is the time, T is the temperature of a point in the body represented by the coördinates x, y, and z, dT/dt is the rate at which this temperature changes, and K, , and c represent physical quantities for the material of which the body is composed, namely, the heat conductivity, density, and specific heat. The combination of these constants in the form is convenient. h is called the thermal diffusivity of the material.

In one dimension, Eq. 8 takes the form

. (9)

If dT/dt equals zero, and if we integrate once, we get the equation which represents the steady-state problem:

. (10)

From physical considerations, the integration constant M is seen to have the meaning

and , (11)

which is the same as Eq. 1.

A more general form of Eq. 8 includes an added term to take account of energy transformations associated with a change of state, and so forth, which will not be considered here.

There are infinitely many solutions to the fundamental differental equation, Eq. 8. Those which are appropriate for a given problem usually comprise an infinite series, the sum of which conforms to the requirements of the geometry of the body, and to the so-called boundary conditions set forth in the problem. The mathematical procedures involved in getting the series required for a particular problem were originally developed by Fourier over a hundred years ago; and these procedures have been extended by other mathematicians to include a great variety of more or less complicated cases.² Here, without taking up the mathematical procedures involved, we will discuss the results of their application to some typical heat problems.³

The infinite slab. First, let us determine the temperature at various points in a plane-parallel slab which, to start with, is at a uniform temperature T₀. We will find the temperature at various places in the slab as a function of the time which elapses after the slab has been immersed into an environment maintained at a fixed temperature T₁. We will assume that T₁ is lower than T₀. (The changes required to apply the results so obtained for the opposite case, in which T₁ is higher than T₀, are obvious.) Practically, if the extension of the slab is great compared to its thickness, this becomes a one-dimensional problem, and to describe it we will take a Cartesian coördinate system which is oriented so that the faces of the slab coincide with the planes x = + x₀ and–x₀.

The solution of Eq. 8, which we want, is a series, the terms of which depend on both x and t. The sum of the series yields a uniform temperature throughout the slab at t = 0; and also at all times it gives a temperature gradient at the surfaces which conforms to the requirements of Newton's law of cooling.

Newton's law of cooling states that the heat lost per unit area of surface, W, by the slab to its environment, is proportional to the difference between the surface temperature T and the temperature of the bulk of the medium in which it is immersed, T₁:

W = N(T–T₁) calories/sec./cm². (12)

W may be resolved into heat lost by radiation, , and heat lost by convection, . The temperature gradient at the surface is determined by the value of W and the thermal conductivity of the material of which the slab is composed.

Stated algebraically, the boundary conditions which our solution of Eq. 8 must satisfy are

at ; throughout the slab (13)

and, for all values of t;

at , (14a)

and also at

x =–x₀ (14b)

The solution of Eq. 8 which satisfies these conditions is

, (15)

where the 's are roots of the trancedental equation

. (16)

The values of may be determined graphically from the intersection points of the two functions of

,

and (17)

Before discussing various aspects of this solution, let us make the substitution,

(18)

in the exponential terms. is called the relaxation time. The reason for this will appear presently.

At the beginning, that is, when t has small values compared with , the accurate expression for T requires several terms of the series given by Eq. 15, in spite of the fact that the series is a rapidly converging one. However, soon after , all the exponential terms become insignificant except the first one (n = 1). This is because is smaller than the other values of . Soon after , Eq. 15 reduces to

(19)

The first factor in the brackets is a constant, the second determines the decay of the temperature difference (T–T₁), and the third factor is the space distribution function for the temperature. The relaxation time is evidently the interval required for the temperature, initially uniform, to assume approximately the distribution given by the last factor in Eq. 19.

The value of for a bo~ly (with vertical sides) in air at room temperature is obtained from Eqs. 35, 48, 12, and 16:

calorie/sec./cm². (35a)

calorie/sec./cm². (48a)

Thus

calorie/sec./cm²/C.

To illustrate how Eq. 19 may be applied, let us consider the case of a telescope mirror of 2 cm thickness which is to be tested by the Foucault knife-edge test. For a reliable test, if this mirror is brought from a room in which it is either warmer or cooler than the air of the testing room, it will be necessary to wait until the mirror has adjusted itself to the new temperature. If the glass is 15 cm or more in diameter and is exposed to the room air on both sides, we may regard it as an infinite slab and apply Eq. 19 to determine its thermal behavior. For the glass we may take K = .0024 and = .0057. This gives = 71 seconds, and by means of Eq. 16, we get =0.219.

Substituting this value of , Eq. 19 can be written in the form

. (20)

This solution is valid after more than 71 seconds have elapsed. To get the thermal behavior at the start, the logarithm of can be plotted as ordinate against t as abscissa. The series of parallel straight lines obtained for t > 71 seconds are then extrapolated to the common point where the abscissa and the ordinate are equal to zero, bearing in mind that , changes rapidly with time when t = 0 and changes very slowly. This method is not very precise, and a more exact solution is to be obtained from Eq. 15. This formula is rather difficult to manage, except in special cases. Two of these are treated below.

Eq. 15 can be simplified for the extreme cases of relatively fast cooling, where /K> > 1, and relatwely slow cooling, where / K < < 1. In the first case is approximately (2n + 1), sinis (–l), and the expression for temperature simplifies to

(21)

For slow cooling, where / K < < 1, the slab is practically isothermal, and the temperature is given by

(22)

The solution of problems of this character will be useful to the experimenter when he encounters questions of design involving the accommodation of objects to changes of temperature.

The application to optical testing has already been discussed. In optical testing with the Foucault knife-edge test, lack of thermal equilibrium distorts the figure of an optical surface and gives rise to troublesome convection currents.

The relaxation time. The relaxation time for a cylinder is approximately half that for a slab, when , the thickness of the slab, and , the diameter of the cylinder, are equal. The relaxation time for a sphere or cube is approximately one quarter of that for a corresponding slab. In most of the nonsteady-state problems encountered, it is sufficient to know the relaxation time. The relaxation time can be interpreted as the time for a heat pulse to travel into the center of the slab, a distance . The relaxation times are given in Table I for a slab thickness of 2 cm ( = 1 cm) for different materials. It must be remembered that for different values of the time required for the heat to penetrate to the center of the slab is proportional to ².

The relaxation time for graphite, which is approximately the same as that for copper, is especially noteworthy. The extreme values for given in Table I are about second and 404 seconds for silver and paraffin respectively.

Periodic temperatures. Let us consider a slab of thickness having a harmonic surface temperature . If is the relaxation time for the slab, the interior temperature is given by the expression

. (23)

The exponential term can be neglected after the relaxation time, and the temperature is then given by the summation. Unless > > 1, the convergence of the series is rapid enough to make the first term a good approximation for it:

. (24)

The product is the ratio of the relaxation time to the period of the impressed harmonic temperature multiplied by 2. If is small, the plate follows the impressed temperature closely with an out-of-phase component, sin , proportional to, and the amplitude of the temperature fluctuation is proportional to .

When > > 17 the temperature near the surface is approximately the same as if the slab were infinitely thick, while the temperature in the center is practically constant.

The temperature at a distance x from the surface of an infinitely thick slab is given, after a long time, by the expression

, (25)

where A cos represents the surface temperature. Thus, the amplitude decreases exponentially with depth according

to the law . There is a time lag of in its harmonic variation, relative to the phase of the surface temperature.

The sphere. When a sphere or cylinder that is initially at a uniform temperature T₀ is introduced into a medium at a lower temperature T₁, the equations similar to those for the slab are:

For a sphere of radius r₀,

(26)

where are the roots of

, (27)

and after the relaxation time the term representing the temperature distribution (corresponding to in Eq. 15) is

.

When,the temperature is given approximately by the expression

(28)

When > > 1, that is, for relatively fast cooling, the temperature is given approximately by

(29)

In the case of slow cooling, in which < < 1, the temperature is sensibly the same throughout the sphere, and its change with time is given by the expression

. (30)

The cylinder. For a cylinder of radius , the temperature in terms of the well-tabulated Bessel functions and is

, (31)

where the 's are roots of the equation

In limiting cases the disappear. For example, when > > 1, that is, when we have fast cooling, the 's are close to the roots of , and the temperature is given approximately by the equation.

. (32)

Temperatures at the center are obtained without the help of tables of Bessel functions because J(0) = 1.

When < < 1, with practically uniform temperature throughout the cylinder,

(33)

If 1, the first term of Eq. 31 dominates after the relaxation time. The Bessel function can be expanded, and then the temperature is given by

(34)

Heat transfer by free convection. Except for a few special cases, the estimation of heat loss by free convection is quite complicated or even impossible. The special cases which have been solved include plane surfaces and wires cooled by convection. The work on this subject up to 1933 has been summarized by W. J. King.⁴ Among the various methods for calculating convection losses, that of Langmnir is the simplest.⁵ His method applies when the surfaces are small, such as those encountered in the laboratory. As it applies to a vertical surface, his method consists of calculating the heat conduction through a postulated stagnant air film of 0.45 cm thickness, thus:

calories/sec./cm². (36)

Here K is the thermal conductivity for air, T₁ is the absolute temperature of the vertical surface, and is the ambient temperature. A more complete theory shows that W is proportional to and to the fourth root of the height of the vertical surface. K, in Eq. 35, is not independent of temperature, and, except for small temperature drops, the heat transfer is given by the expression

calories/sec./cm². (36)

Values of for air are given in Table II to facilitate calculation. These values are defined by the expression

. (37)

Langmuir found that heat losses by free convection from a horizontal surface facing upward are 10 per cent greater than they are from a vertical surface, and they are 50 per cent less from a surface facing downward than they are from a vertical surface.

The procedure for calculating the convection losses from wires is also treated by Langmuir.

Heat transfer by radiation. The energy emitted by a surface of area A radiating the heat spectrum between the wave lengths and is

calories/sec. (38)

This represents the summation of energy in respect to the solid angle over the hemisphere (angle 2 steradians). Here is the emissivity of the surface. This is the ratio of the emission of the surface to that which would obtain for a "black body" at the same temperature. is the energy radiated per unit solid angle by a black body of the same area at wave length for a unit wave-length range, cm.

So-called black-body radiation is defined as the thermal radiation coming from the surface of a body which is in temperature equilibrium with all of its surroundings. For example, the inner surface of a cavity in an opaque material at a uniform temperature emits black-body radiation. In fact, black-body radiation is obtained experimentally from just such a cavity. The wall of the cavity is pierced to form a small aperture to serve as an outlet for the radiation, the hole being small enough not to disturb the equilibrium perceptibly. The name black-body radiation comes indirectly from Kirchoff's law, which states that the emission and absorption coefficients of any body are equal. A black body with an absorption coefficient of unity, , therefore, by Kirchoff's law, has an emission coefficient of unity, = 1.

The Planck expression for is a function of the wave length, , and the absolute temperature, T.

calories/sec./cm/steradian. (39)

For this expression is approximated to within 1 per cent by the so-called Wiens formula,

calories/sec./cm/steradian (40)

As becomes < < 0.3, , becomes asymptotic to .

For = 80 the expression is approximated to within 1 per cent by the so-called Rayleigh-Jeans formula,

calories/sec./cm/steradian. (41)

As becomes > > 80, becomes asvvmptotic to . The values of the constants and , where is expressed in centimeters, are calorie/sec./cm²/unit solid angle; and = 1.432 cm degrees.

The total heat lost by a unit area of the surface of a "black body " is the quantity expressed by Eq. 38 integrated over all wave lengths. This gives Stefan's formula:

calories/sec. (42)

Most surfaces have a total emission which may be expressed as

calories/sec. (43)

Here has the value of 1.38 X 10 calorie/sec./cm/ degree⁴.

The heat emitted by a flat surface of area A into a cone which is defined by a solid angle is

calones/sec. (44)

Here is the projected area of the source and is the fraction of the total heat emitted in the direction defined by the element .

is an emissivity averaged over all wave lengths, and it is ordinarily "constant" only for a small temperature range. For porous nonmetallic substances it is very nearly unity, regardless of the color of the material. Naturally, the visible color of a body does not determine its infrared "color." Some bodies, such as white lead, are almost completely black throughout the heat spectrum, while the reverse is true for other substances, notably soot and black paper, both of which are transparent for the long wave-length end of the heat spectrum. for aluminum paints, around room temperature, varies between 0.3 and 0.5. For nonmetallic pigment paints = 1.

For clean metals, varies with the temperature in such a way that the total emissivity is conveniently represented by an expression of the form

calories/sec. (45)

Here M and m are constants. Values of M and m for some common metals are given in Table III.

The heat transfer by radiation between two parallel black isothermal surfaces of area A at absolute temperatures and , which are separated by a small distance, is

calories/sec./cm². (46)

calories/sec./cm². (47)

If the temperature difference, (-), is small, this heat transfer may be expressed so:

calories/sec./cm². (48)

Thus, owing to the fact that the absolute temperature enters the expression to the third power, we see that the importance of radiation as an agency for heat transfer becomes greater at higher temperatures, until finally, in comparison, ordinary conduction becomes negligible.

At extremely high temperatures, the action of an insulator is the same as the action of a radiation baffle or series of baffles. The effect of baffles can be illustrated by the example of two infinite plane-parallel black surfaces at temperature and . If a thin black baffle is interposed between these two surfaces, the transfer by radiation is reduced to one-half its original value. Two baffles reduce it to one-third, three baffles to one-fourth, and so forth, and if, instead of black baffles, polished metal reflectors are used, the insulation effect is even greater. In high-temperature furnaces the furnace tube with its winding is frequently surrounded by a thin sheet of some metal like molybdenum to serve as a baffle to reflect back most of the radiant energy emitted by the tube and so to decrease the power required. Sometimes, too, a second refractory furnace tube may surround the first to act as an insulator.

Low temperatures. Moderately low temperatures are obtained in the laboratory by immersion in baths of ice, salt and ice, dry ice, liquid air, and so forth. The various temperatures so attained are listed in Table IV. For obtaining extremely low temperatures, the methods required are very elaborate.⁶

Methods of obtainmg high temperatures. Flames. The use of flames affords the most simple and convenient means Fig. 4 illustrates the use of a blowpipe wit'n the alcohol lamp, showing how the cheeks are used as bellows to give continuous air pressure.

Fig. 5 illustrates the ordinary Bunsen burner. The Bunsen burner is simply a tube arranged with a fuel gas jet in' the bottom and air ports in thei sides near the bottom. It draws air in through these ports by injector action of the gas jet. The air drawn into the tube at the bottom is mixed with the fuel gas as it passes up through the tube; and above the top of the tube this air reacts chemically with the gas fuel to produce the flame.

The Bunsen burner draws only about half as much oxygen more air were mixed with the gas, the velocity of propagation of the flame would be greater than the upward velocity of the gas in the tube, and the flame would "backfire." However, additional air required for combustion of the gas is supplied to the flame above the burner tube; owing to the more abundant supply of air, at the edges of the flame the propagation velocity is greater than the upward velocity of the gas, so that the fire does not blow itself out.

Natural gas, which contains less hydrogen than coal gas, has a much smaller flame velocity. (The heat of combustion and the chemical composition of some commercial gases are shown in Table V.) Accordingly, with natural gas there is a greater tendency of the burner to blow out than with coal gas. This has resulted in the invention of fixtures like those shown in Fig. 5(b) and (c), which serve to retard the upward velocity of a portion of the gas mixture. The flame formed by this slowed-up portion does not blow out, and it prevents the main flame from doing so. A small tube may be soldered to the burner as shown in Fig. 5 at (a) to act as a pilot as well as to prevent the flame from blowing out.

The Meker burner is a Bunsen-type burner with the top of the burner tube flared out and fitted with a nickel grill. This is illustrated in Fig. 6. The Meker burner can burn coal gas with a higher air admixture than the Bunsen burner, because the grill, acting as a Davy lamp screen, prevents the flame from backfiring. The hot inner blue cone of the Bunsen flame is replaced here by an array of small cones, one over each element of the grill. This array produces a flame which is both hotter and more uniform over an extended area than the Bunsen flame. The temperature distributions in the Bunsen and Meker flames, with coal gas fuel, are shown in Fig. 7.

To obtain a higher temperature than either the Bunsen or Meker will yield, the fuel is burned with air or oxygen under pressure at the end of an orifice with burners such as the ones shown in Figs. 8 and 9. When natural gas is burned with air, a special tube end is required. (See Fig. 9.) Another method of burning gas to get a high temperature is to project a jet of air or oxygen through a gas flame as shown in Fig. 10. A simple method using a water aspirator for obtaining compressed at moderate pressures is shown in Fig. 11.

Extremely high temperatures are attained with oxyhydrogen or oxyacetylene torches. Commercial torches like the one illustrated in Fig. 12 are recommended for these fuels.⁷ These torches are equipped with a mixer, usually in the handle, to produce a homogeneous solution of the fuel and oxygen gases. It is very important to have such a homogeneous mixture of oxygen and fuel; otherwise the torches would blow themselves out. The type of orifice used is illustrated in Fig. 13. Fig. 14 shows the distribution of temperature in the oxyacetylene flame and also in the carbon arc.⁸

A furnace is required to heat objects to higher temperatures than those that are obtainable with torches. Gas furnaces for use in the laboratory are shown in Figs. 15 and 16.

Oxygen-gas furnaces can be made to yield very high temperatures; for example, Podszus and von Wartenburg, Linde, and Jung have described furnaces with a zirconium dioxide tube using illuminating gas or oil vapor as fuel.⁹ These furnaces attain temperatures of about 2600 C.¹⁰

Electric furnaces. Electric furnaces for temperatures to 500 C., useful for such applications as the baking out of charcoal traps, can be made by winding a coil of Nichrome or Chromel wire on an iron tube as shown in Fig. 17. The tube is first covered with a piece of mica or asbestos sheet to avoid shorting out the winding. A sirnple way of fastening the ends of the winding is illustrated in Fig. 18. Various types of insulation may be used. For example, the inner tube and its resistance wire winding may be covered with several layers of asbestos. The furnace is assembled with transite¹¹ ends, using Insa-lute cement. It is necessary to avoid contact between the Insa-lute cement and the furnace wire at elevated temperatures.

Nickel wire is suitable for a furnace winding. However, its resistance changes approximately twofold when it is heated from room temperature to 500C. This behavior is in contrast to the behavior whose change of resistance is negligible. The change of resistance of nickel may or may not be desirable; it may be desirable to have a large coefficient if the resistance is to be used for regulating the temperature of the furnace.

Electric furnaees whieh operate in air to 1100C. may be made with the niekelehrome alloys as resistors, a poreelain, Alundum, quartz, or magnesia tube being used to support the winding. Diatomaceous earth makes an excellent insulator.¹² A useful furnace construction for the laboratory is illustrated in Fig. 19.

Platinum may be used as resistor for temperatures greater than 1100C., when it is desired to have the furnace operate in air. This resistor will operate up to a temperature limit of 1600C. In order to obtain a furnace temperature as near this limit as possible, Orton and Krehbiel used a Chromel "booster" winding on a tube mounted outside of and concentric with the platinum winding.¹³ Theplatinumwire may be wound on quartz glass, which has a temperature limit in air of 1300C., on unglazed porcelain, for which the limit is 1400C., or on clay, with a limit of 1700C. However, best of all is an Alundum tube (alumina with clay binder). Its limit, 1900C., is above that of the platinum.

Silicon is formed from quartz or porcelain in a reducing atmosphere, and silicon attacks platinum. Accordingly, it is best to use a platinum-wound furnace in an oxidizing atmosphere. If, however, the wire is wound on an Alundum tube, it may be operated in a reducing atmosphere.

Molybdenum or tungsten can be used as a resistor in an atmosphere of hydrogen; the limiting temperatures attainable are 2200C. and 3000C. respectively. As a support for the resistor winding, Alundum can be used to 1900C., magnesia to 2200C., zirconia to 2500C., and thoria to 3000C. Porcelain is unsuitable, for the reason given above; namely, hydrogen blackens it at high temperatures.¹⁴ A tungsten (or molybdenum) furnace is shown in Fig. 20. Most refractories cannot be subjected to high temperatures in vacuum because they either evaporate or are reduced by the vacuum (oxygen formed by dissociation is pumped away).

Carbon and graphite tube furnaces can be operated to a temperature of 2000C. in vacuum. Above this temperature the carbon begins to vaporize, the carbon begins to vaporize, and at 2500 C. the rate of evaporation is rapid. In hydrogen or nitrogen the temperature limit is 2000C. At this limit chemical action between the carbon and the gas setsin. However,inanatmosphere of carbon monoxide, carbon resistors may be used at temperatures over 3000C. A f urnace designed byArsem,¹⁵ which may be.operated either in an atmosphere of carbon monoxlae or in vacuum, is shown in Fig. 21. This furnace has its resistor tube cut into a spiral to increase its resistance'. and flexibility. Connections are made to the ends of the resistor tube with water-cooled copper Jaws.

Carbon grain resistors such as the one shown in Fig. 22 have a higher electrical resistance than solid carbon and are often useful in the laboratory.

A carbon-arc furnace is shown in Fig. 23.

Fig. 24 shows apparatus used for melting metals in vacuum by heating with high-frequency current. It is peculiar to this method that the metal charge is at a higher temperature than the crucible, a fact of practical value when working with extremely refractory metals.¹⁶

Fixed temperatures. Constant temperature may be maintained at 0C. with melting ice, and at the boiling temperature of water by means of a device such as the one illustrated in Fig. 25. Other liquids and solids may be used for maintaining other constant temperatures; for example, a temperature of 444.6C. is obtained by boiling sulphur. Some of the fixed temperatures useful for the calibration of thermometers and thermocouples are given in Table VI.

Let us compare this method with one which employs a thermocouple inside the furnace as a pilot. We see that there will be more lag between the time the heat input is altered and the time it affects the thermopile. As a result, with the thermocouple pilot the limits of the fluctuation of the furnace temperature are separated more than they are when the resistance of the heater wire serves as the pilot.

Even when the furnace heater wire serves as the pilot, there are fluctuations due to the period of auxiliary instruments. These temperature fluctuations may be diminished simply by interposing alternate shells of thermal "ballast" and insulator between the furnace winding and the region that is to be kept at a constant temperature. The temperature diffusion through such alternate shells is slow. The furnace tube itself, which separates the heater wires from the constant temperature region within, is usually adequate for this, because of its relatively low diffusivity, h; for example, one may obtain temperatures constant to about 0.01C.inside the furnace tube even when the period of temperature oscillation of the furnace wiring outside is of the order of 30 seconds.

The device shown in Fig. 26 is convenient for temperature regulation.¹⁸ The two bulbs of this device have equal volumes, and they are equipped with identical nickel-chromium alloy heaters and connected electrically as shown in the diagram, Fig. 27. The bulbs are filled with air and the pressures on either side of the mercury column are such as to hold the top surface of the mercury at the level of thetungsten contact when the voltage drop over the left resistance (see Fig. 26) is the same as the voltage drop over the right resistance. These voltage drops are equal when the temperature-sensitive feeler resistance is the same as the fixed constantan resistance. (See Fig. 27.) These resistances are ad~ justed to be equal at the desired temperature. If the temperature of the feeler resistance is too high or too low, the heating in the two bulbs is unequal, and the resulting change in pressure in the bulbs opens or closes the mercury contact, and this in turn operates a relay actuating the heating and bridge current.¹⁹

The resistance used to operate the regulating device may be either the heater resistance or it may be separate from the heater. In the latter case this arrangement is suitable for maintaining a constant temperature in a room. The feeler resistance is strung back and forth near the ceiling of the room (at about 8 feet above the floor). For such an application the heaters, which the regulating device controls, are situated in front of the ventilator air inlet to the room.

Thermostat baths use water for ordinary temperatures, oil or eutectic salt mixtures for elevated temperatures, and alcohol for low temperatures. Beattie gives the composition of two eutectic baths. (See Table VII.) These baths are useful in the temperature range above 120C. Thelower limits of their temperature ranges overlap the upper temperature limits of mineral-oil baths (150 to 200C.) andheavy cylinder oil baths (150 to300C.).

The temperature of a water bath is controlled by regulating the heat onput. A mercury-in-glass bulb with contacts coupled to a relay as shown in Fig. 18, Chapter X, is suitable for a bath heated electrically. The device shown in Fig. 28 is effective for controlling the temperature of a gas-heated bath. The aperture through which the gas for the flame passes is regulated by the thermal expansion and contraction of the mercury. With these devices the fluctuations of the bath are about 0.1C.

Temperature measurement. Temperature is always measured practically by a measurement of some temperaturesensitive property, such as light emission, electrical resistance, length, volume, thermal e.m.f., and so forth. All physical properties which vary with temperature are possibilities for such a measurement, although some properties, like electron emission, are so strongly influenced by chemical impurities or by the past physical history of the thermometric substance that they are useless.

Liquid-in-a-bulb thermometers depend upon change of volume with temperature for th'eir readings. Among them, two are of unusual interest. One, which was manufactured in Germany at one time, used gallium as liquid and fused quartz for the bulb and capillary. This thermometer was useful up to a temperature of about 1000C., in contrast to the mercury thermometer, which is ordinarily useful only to 200C. However, with a high pressure of nitrogen (up to 40 atmospheres) mercury-in-glass thermometers may be heated considerably above 200C. A graphite thermometer with molten tin as the liquid has been made by Northrup.²⁰ This thermometer may be used to 1680 without chemical reaction between the tin and the graphite. As tin does not boil at 1680C., Northrup thinks that the limit in temperature of this thermometer is probably several hundred degrees higher. The position of the tin in the graphite capillary is determined by a tungsten feeler. For gallium the temperature range from the melting point to the boiling point is from 29.7 to 1600C., and for tin it is from 231.8 to 2260C. The operation of the two thermometers described above depends upon these unusually long temperature intervals between the melting points and boiling points.

Thermocouples operate by virtue of the temperature dependence of the thermal e.m.f. generated by two substances in contact. The thermocouple may be employed in the laboratory for temperature measurement from liquid air temperatures to the melting temperature of molybdenum.

The base metals commercially available are commonly used as thermoelectric wires. Chromel-Alumel wires have a high coefficient of thermal e.m.f. They are obtainable from the factory matched to give the temperature to 5•C. Copper and constantan wires also have a high thermal e.m.f. These wires have the advantage over Chromel-Alumel that they are easily soldered. The LeChatelier combination (platinum and 10 per cent platinum-rhodium) is used for precision measurements. Special thermocouple metals, such as tungsten, molybdenum, and their alloys, are useful at very high temperatures.

It may be desired to calibrate a particular thermocouple with fixed standard-temperature baths such as those listed above in Table VI. The best procedure is to use the calibration curve supplied by the factory, which gives the e.m.f. at frequent temperature intervals, and to plot an empirical correction curve for it from the calibration-data.

Radiation pyrometers determine temperature by the measurement of light emission. There are several types, and descriptions of them and their operating characteristics appear in many books. The type in most common use measures, with a special photometer, the intensity of monochromatic light (6600 A) emitted by the incandescent body whose temperature is being measured.

Table VIII is useful for estimating the temperature of a body from Its color.

² Carslaw, H. S., Introduction to the Mathematical Theory of the Conduction of Heat in Solids, Seeond Edition. London: The Macmillan Company, 1921. Ingersoll, L. R., and Zobel, O. J., The Mathematical Theory of Heat Conduction, With Engineering and Geological Applications. Boston: Ginn and Company, 1913.