Proceedings of the Tomsk Order of the Red Banner of Labor Industrial Institute named after S. M. Kirov

Vol. 59, II. 1941

THE PROCESS OF A STEAM BOILER IN \(I\)-\(t\) COORDINATES

G. I. Fuks

On the basis of the \(I\)-\(t\) diagram it is possible to construct the thermal process of a boiler on the gas side in \(I\)-\(t\) coordinates. Analogously to the representation of the process of a steam turbine in \(i\)-\(s\) coordinates, this method simplifies and refines the calculation.

The construction is based on the following obvious relations. For the entire convective heating surface and any part of it we have, per kilogram of fuel,

\[ Q=c_g V_1 T_1-c_g V_1 T_2-c_v (V_2-V_1)(T_2-t)-Q_5 \]

or

\[ Q=V_1(I_{T_1}-I_{T_2})-(V_2-V_1)(i_{T_2}-i_t)-Q_5. \tag{1} \]

The second term on the right-hand side represents the heat expended in heating the intake air from the temperature of the intake air \(t\). The quantities \(I\) and \(i\) are the heat contents (enthalpies) of the gases and of the air, respectively, at the temperature given as the subscript.

A simple transformation of this dependence leads to the relation

\[ Q=V_1 I_1 - V_2 I_2 + (V_2-V_1)i_t - Q_5. \tag{2} \]

On the other hand, the heat balance of the installation gives

\[ Q_n^p=Q_l+Q_k+Q_5^m+Q_5^k+Q_2+Q_3+Q_4, \tag{3} \]

where \(Q_l\) and \(Q_k\) are the heat transferred by radiation and convection,
\(Q_5^m\) and \(Q_5^k\) are the heat lost to radiation by the furnace and by the convective part.

Further, obviously,

\[ \eta_y=\frac{Q_l+Q_k}{Q_n^p}. \tag{4} \]

In the presence of an air preheater, \(Q_l\) includes the heat transferred to the air in the air preheater, but \(Q_5^{\mathrm{vp}}\) of the air preheater must be included in the quantity \(Q_5^k\).

Combining (1) and (3) for the entire boiler installation, we obtain

\[ Q_n^p=V_m(I_{T_{\mathrm{sh}}}-I_{T_y})-(V_y-V_m)(i_{T_y}-i_t)+Q_l+Q_5^m+Q_2+Q_3+Q_4. \tag{5} \]

Using this equation, we carry out the following construction. Having read from the \(I\)-\(t\) diagram\(^1\) the values of \(I\) according to the composition of the gases in the furnace and multiplying these values by \(V_m\), we plot \(V_m I\) as a function of \(T\) (Fig. 1). In the same way we plot the curve \(V_y I\).

\(^1\) See G. I. Fuks, “On determining the composition and heat content of flue gases in the calculation of a boiler installation,” Soviet Boiler-and-Turbine Engineering, No. 7, 1937.

According to the well-known formula

\[ Q_{2}=V_{y}(I_{TV}-I_{t}) \tag{6} \]

we lay off \(Q_{2}\). Next, on the ordinate corresponding to \(T_{y}\), the heat balance of the installation is constructed according to (5) (Fig. 1).

According to the drawing, it is then possible to lay off \(Q_{n}\), \(Q_{\ell}\), and the theoretical combustion temperature \(T_{2}\). Subsequently it remains only to divide the convective heat \(Q_{k}\) among the individual heating surfaces.

The indicated construction is final if there is no air heater in the installation. If the latter is present, the fraction of heat used on the convective heating surface of the boiler will decrease and, correspondingly, the fraction of radiant heat will increase. The corresponding rearrangement of the installation balance is carried out on the abscissa axis (Fig. 1).

Fig. 1.

Having calculated the heat used in the air heater:

\[ Q_{o}=\alpha_{m}V_{0}(i_{tв'}-i_{t}), \]

we add it to \(Q_{\ell}\) and subtract it from \(Q_{k}\) (Fig. 1). This makes it possible to lay off the actual values

\[ Q'_{k},\ Q'_{\ell}\ \text{and}\ T'_{2}. \]

After determining the indicated quantities, we proceed to plotting the boiler process. For this purpose, in the same coordinates we construct the curves \(VI=f(T)\), taking the composition of the gases and the volume at the characteristic points with account taken of air leakage. In Fig. 2 the curves are plotted with account taken of the volume and composition of the gases at 3 characteristic points: before the superheater (curve \(A\)), after the superheater (curve \(B\)), and after the second surface (curve \(C\)).

We plot the further subdivision of the convective heat among the individual surfaces using relation (2). Thus, for the 1st surface, starting from point 1 on the curve \(V_{m}I\), we lay off to the right the segment \((V'_{n}-V_{m})i_{t}\) and then, to the left, the segment \(Q'_{5}\) for this surface. If, for example, the temper-

the gas temperature before the superheater \(T'_n\), then the segment \(Q^k_1\)—the heat used on the 1st convection surface—is simply determined graphically. For the superheater, usually in the calculation one can determine it from the heat balance, \(Q^n_n\)—the heat used in the superheater per kg of fuel. Then the gas temperature after the superheater, \(T''_{nn}\), etc., is immediately read off.

Of course, with such a construction one should obtain

\[ Q^k_1 + Q^{nn}_n + Q^k_2 + \ldots = Q'_k \]

and

\[ Q'_5 + Q^{nn}_5 + Q''_5 + \ldots = Q^k_5 . \]

Such a construction must be carried out for all boiler surfaces (including the water economizer, if it is present in the installation) up to the air heater.

Drawing 2.

In a practical calculation both drawings (1 and 2) are combined into one. The merits of the proposed method should be considered to be:

1. An accuracy of calculating temperature differences not inferior to the analytical method of calculation.
2. Clarity and simplicity.
3. Constant self-checking at all stages, since verification of the balance occurs automatically. Of course, if, for plotting all the curves, the quantities \(l\) have to be calculated analytically, the calculation will become very laborious. This circumstance is avoided by using the author’s indicated diagram.

For “normal” types of boilers the entire calculation is conveniently combined in the following table.

Table No. 1