# Curves for the Sensible Heat Capacity of Furnace Gases

Table of Contents

Knowledge of the thermal capacity of gases is of great importance in making metallurgical calculations. The metallurgist is frequently called upon to investigate and determine furnace efficiencies in which the heat carried into or out of the furnace by gases is a large item in the heat balance. Not only do such problems present themselves in the determination of furnace efficiency, but also in the study of the application of heat is accessory apparatus, such as stoves, regenerators, waste-heat boilers, driers, etc. The thermal effect of the use of excess air in the combustion of fuel, the theoretical temperatures of combustion, the quantity of heat in hot blast at various temperatures, the effect of hot blast on furnace temperatures, are a few more examples of frequently occurring calorific problems involving gases. So many are the applications of the data on the heat capacity of gases that the subject merits careful study.

The heat in a gas maybe due to its heat of combustion, if it is a combustible gas, and to its temperature. The latter is called its sensible heat and is the heat absorbed or evolved by a gas as its temperature is raised or lowered. The heat of combustion of gases is well established and is commonly known. The values can be found in almost any modern treatise on metallurgy. The sensible heat of gases has not been so well established because it is in most cases a function of the temperature, and the values of specific heats of gases over a wide range of temperatures have only recently been determined. It is the purpose of this paper to deal with the sensible heat of gases.

The calculation of sensible-heat capacities from specific-heat equations is a comparatively long and tedious operation. This is especially

true in the case of polyatomic gases the specific-heat equations of which are quadratics or are higher in degree.

To simplify this calculation the authors have selected what have appeared to be the most reliable and recent data and have constructed a series of curves by means of which the sensible heat in a gas at any temperature or between any temperatures up to 2,000° C., or 3,600° F., may be easily determined. The use of these curves has not only affected a great saving in time, but has encouraged the making of many calculations which would not have been attempted if the more laborious method was used.

## Experimental Data on the Specific Heat of Gases

Dr. Joseph W. Richards has, in his Metallurgical Calculations, Part I, given formulae for the mean specific heats of all the common gases in terms of both the imperial and metric systems of measurement. These formulae are derived from the results of experimental determination of the specific heats at various temperatures. The data were obtained from the researches of Regnault, Mallard, and LeChatelier. Since the time of publication of that book, however, other investigators have been at work on the subject. Among these are: Pier, Holborn, Austin, Henning, Njerrum, Furstenau, and Swan. A summary of the results of their researches has been very ably made by Messrs. Lewis and Randall, who presented their conclusions in a paper entitled A Summary of the Specific Heats of Gases. They have selected formulae for molal heats of gases to agree as closely as possible with all the experimental work, and where a difference in data existed the values obtained by the most reliable method were chosen. Since their formulae agree with the most probable values of all the experimental data, they are especially valuable, and have been used in constructing the curves for heat capacity that are the subject of this paper.

## Specific-Heat Equations & Derivation

As noted above, Lewis and Randall give formulae for the molal heats of the gases. It has been necessary to convert these formulae into more convenient form for use with either volumetric or gravimetric units in either the Imperial or Metric system.

To facilitate the use of formulae, the following symbols have been adopted:

Cp = molal heat = heat required to raise the temperature of a molecular weight of the gas 1° under constant pressure.
cp = specific heat under constant pressure.
Cm(t1 to t) = mean specific heat between the temperatures t1 and t.
T = absolute temperature, degrees centigrade,
t = temperature, degrees centigrade.
t = temperature,, degrees Fahrenheit.
Q(t1 to t) = quantity of sensible heat (heat capacity) between the temperatures t1 and t.

The following formulae were selected by Lewis and Randall:

Nitrogen, oxygen, carbon monoxide, Cp = 6.50 – 0.0010T
Hydrogen, Cp = 6.50 – 0.0009T
Water vapor, hydrogen sulphide, Cp = 8.81 – 0.0019T + 0.00000222T²
Carbon dioxide, sulphur dioxide, Cp = 7.0 + 0.0071T – 0.00000186T²

Table I gives the formulae for mean specific heats that we have calculated from the above molal heats for 1 kg.,1 cu. m., 1 lb., and 1 cu. ft. of each gas.

The values for methane are calculated from a formula given by Fulton (Principles of Metallurgy, p. 408) on the authority of J. V. Ehrenwerth (Metallurgie, vol. vi, p. 306).

For an explanation of the method of converting specific-heat formulae to mean specific-heat formulae see Appendix A.

## Heat-Capacity Curves

### Construction and General Application

The amount of sensible heat that a gas will absorb between any two temperatures t1 and t is equal to the product of the mean specific heat between those two temperatures times the difference in temperature. Therefore, if one temperature is zero,

Q(0 to t) = cm t

By substituting in this equation the various values of cm given in the preceding tabulation and then substituting various values of t, any desired number of values of Q may be obtained. From these values a curve can easily be plotted. Such a curve will give for any temperature up to 2,000° C., or 3,600° F., the sensible-heat capacity of the gas.

On Plate I we have drawn curves for each gas giving the sensible-heat capacity in British thermal units per pound and also per cubic foot (measured under standard conditions). On Plate II are corresponding curves giving kilogram-calories per kilogram and per cubic meter (also measured under standard conditions).

It should be noted that the values given apply to the substances only in the gaseous state.

Dissociation is not great within the range of temperatures for which the curves are drawn. (See Fulton, Principles of Metallurgy, p. 414.)

The general application of the curves is simple. The temperatures are laid off along the axis of ordinates and the sensible-heat capacities are laid off along the axis of abscissae. If the temperature of a gas is known and it is desired to find the sensible-heat capacity between zero and this temperature, one reading gives it directly. If two temperatures are given and the sensible-heat capacity between these two temperatures is desired, two readings must be taken and subtracted one from the other.

In addition to Plate I and Plate II we have drawn two more series of curves in Plates III and IV, which give the actual specific heats at various temperatures, the former for units of weight and the latter for units of volume. These are for use in finding the theoretical temperature of combustion, as explained in Appendix B and as illustrated in Example 4.

## Application in Metallurgical Calculations

The applications of the curves to metallurgical problems are many. On account of lack of space only four typical examples, just enough to illustrate the use of the curves, will be worked out. Special attention is directed to Example 4, which illustrates a method of finding the theoretical temperature of combustion much more easily than by any other method we know of.

### Example 1

A reverberatory furnace discharges 10,000 cu. ft. of waste gases per minute (measured under standard conditions) at 2,000° F. The volumetric analysis of the gas is as follows:

Find the sensible heat in the waste gases above 0°F.

Referring to Plate I, we obtain the following values for each of the constituent gases at 2,000° F.

Sensible-heat capacity, B.t.u. per cubic foot.

Combining these in the proper proportion we get:

49.6 x 0.08 + 63.6 x 0.15 + 40.8 x 0.77 = 44.92 B.t.u. per cubic foot,

which is the sensible heat in 1 cu. ft. of the waste gases. Therefore the total sensible heat passing out of the furnace in the waste gases each minute is 439,500 B.t.u.

### Example 2

Suppose the gases of Example 1 enter a brick regenerator at 2,000° F. and leave it at 600° F., and that the regenerator subsequently heats 8,000 cu. ft. of air from 70° F. to 1,200° F. for every 10,000 cu. ft. of waste gases (both air and gases being measured under standard conditions). What is the calorific efficiency of the regenerator? From Example 1 we know that the sensible heat in the gases at 2,000° F. is 439,500 B.t.u. Proceeding in the same manner (using Plate I), we find that at 600° F. the sensible heat per cubic foot is:

14.1 x 0.08 + 16.2 x 0.15 + 11.6 x 0.77 = 12.49 B.t.u.

The sensible heat in 10,000 cu. ft. at 600° is therefore 124,900 B.t.u. . The heat input of the regenerator is therefore

439,500 – 124,900 = 314,600 B.t.u.

The heat in 1 cu. ft. of air at 1,200° F. is 23.7 B.t.u. and at 70° F. it is 1.3 B.t.u. Therefore, the net heat absorbed by 1 cu. ft. of air is 22.4 B.t.u. and the heat output of the regenerator is 8,000 x 22.4, or 179,200 B.t.u.

∴ Efficiency of regenerator = Heat output/Heat input x 100 = 179,200 x 100/314,600 = 57.0 per cent.

### Example 3

At an iron blast-furnace plant a stove receives 35,000 cu. ft. of blast-furnace gas per minute. The heating value of this gas is 98 B.t.u. per cubic foot and its volumetric composition is:

The gas is at a temperature of 600° F.
The stove discharges 61,000 cu. ft. of gases at 550° F. and of the following volumetric composition.

The temperature of the air that is used for burning the gas in the stove is 50° F.

The stove (while on air) heats 100,000 cu. ft. of air from 50° F. to 1,100° F. for every 35,000 cu. ft. of the original blast-furnace gas.

All volumes are measured under standard conditions of temperature and pressure.

What is the efficiency of the stove?

The per cent, efficiency is Heat output/Heat input(net) x 100

### Example 4

Find the theoretical temperature of combustion, or “pyrometric effect,” of the blast-furnace gas in Example 3.

The total sensible-heat capacity of the 61,000 cu. ft. of the products of combustion is 3,989,880 B.t.u., or 65.4 B.t.u. per cubic foot.

The composition of the products of combustion is:

The theoretical temperature of combustion t is found by the formula:

in which, t1, t2, t3, etc., are the theoretical temperatures of combustion of each of the constituent gases, if we consider for a moment that the gases contain only that constituent and no other. In other words t1, t2, t3, etc., are the temperatures corresponding to the points of intersection of the curves on Plate I with an ordinate drawn through 65.4 B.t.u.

P1, P2, P3, etc., are equal to the volumetric percentage of each constituent gas.

C1, C2, C3, etc., are equal to the mean specific heat of 1 cu. ft. of each constituent gas between t and t1, t and t2, etc., and are obtained by reading the specific heats from the curves of Plate III just above or below the temperatures t1, t2, t3, etc., so that the readings will be very closely the mean specific heats between t and t1, t and t2, etc.

For a full explanation of the method of obtaining the formula for the theoretical temperature see Appendix B.

In this particular example we shall let the subscripts 1, 2, 3, and 4 represent respectively, CO2, H2O, O2, and N2.

From Plate I, we obtain by erecting an ordinate through 65.4 B.t.u.

t1 = 2,045° F., t2 = 2,520° F., t3 and t4 = 3,075° F.

From Plate III, we obtain:

c1 = 0.0385, c2 = 0.0335, c3 and c4 = 0.0235

From the analysis:

p1 = 22.1, p2 = 4.3, p3 = 2.0, p4 = 71.6

Substituting: t = (2,045 x 22.1 x 0.0385) + (2,520 x 4.3 x 0.0335) + (2.0 + 71.6) ( 3,075 x 0.0235)/(22.1 x 0.0335) + (4.3 x 0.0335) + (73.6 x 0.0235) = 7,423/2.725 = 2,724° F.

## Appendix A

Method of Transforming the Equation for Specific Heat into the Equation for Mean Specific Heat

Let the curve in Fig. 1 represent the general equation for specific heat,

cp = A + BT + CT² + DT³ + etc.

In which A, B, C, D, etc., are constants and T is the temperature.

Let T1 and T2 be any two temperatures on the curve.

The mean specific heat cm between T1 and T2 will therefore be,

Substituting the general equation for c,

## Appendix B

### Derivation of Formula Theoretical Temperature of Combustion

Consider the products of combustion of a combustible gas as made up of several constituent gases, which we shall call Gas 1, Gas 2, Gas 3, etc.

In Fig. 2, the curves represent the sensible-heat capacity curves for the series of constituent gases, Gas 1, Gas 2, Gas 3, etc.

Q = sensible-heat capacity of the combined gases of combustion from 0° to t°.

Q1, Q2, Q3, etc. = ditto for Gases 1, 2, 3, etc., from 0° to t1, t2, t3, etc.
t = theoretical temperature of combustion of the gas t1, t2, t3, etc.
= theoretical temperature of combustion if the total product of combustion were only one gas, Gas 1, Gas 2, etc.

P1, P2, P3, etc. = portions of Gas 1, Gas 2, Gas 3, etc., in the products of combustion.

Let m1, m2, m3 = average slopes of curves connecting the points Q1 and t1, Q2 and t2, etc.