Student K. P. Schmidt

K. P. Schmidt’s Graphical Method as Applied to the Calculation of a Steam-Locomotive Boiler

In my preceding article I set forth the theoretical foundations of a new graphical method for carrying out heat calculations in steam boilers. Its practical application to the calculation of a steam-locomotive boiler is of great importance, greatly simplifying and facilitating the work of the designer. In order that everyone wishing to make use of this calculation may have all the details clear, I have given below a concrete example of the heat calculation of a steam-locomotive boiler.

To reduce the arithmetical computations, we shall make use of the data of the approximate heat calculation of a locomotive, given on pp. 244–246 of Prof. Syromyatnikov’s book The Heat Process of the Steam Locomotive.

Using this source, we obtain the following initial data: \(M = 3122\), \(N = 0.3345\); the coefficient of division of the gases between the fire tubes and the smoke tubes \(\beta = 0.59\); combustion temperature \(T_0 = 1617^\circ C\), temperature at the rear tube plate \(T_1 = 992^\circ C\); heating surface of the smoke tubes \(H_g = 86.7\ \mathrm{m^2}\), the same for the fire tubes up to the superheater \(H_{\text{ж}'} = 6.4\ \mathrm{m^2}\); the same along the length of the superheater elements \(H_{\text{ж}''} = 60.2\ \mathrm{m^2}\); heating surface of the superheater \(H_{\text{п}} = 65.0\ \mathrm{m^2}\); sum of the cross-sectional areas of the smoke tubes \(\Omega_g = 0.199\ \mathrm{m^2}\).

As has already been mentioned, the heat-transfer equation is most correct with the exponent \(n = 4/3\). Moreover, with this exponent Prof. Syromyatnikov gives the value of the heat-transfer coefficient as a function of the “mass velocity” (see Syromyatnikov, The Heat Process of the Steam Locomotive, p. 71), i.e., of the product \(\gamma w\)—the density of the combustion gases times their velocity. Since the velocity of the gases in the tubes is the greater the smaller their density, it is not difficult to see that this product is simply equal to the quotient obtained by dividing the weight quantity per second of gases passing through the tubes by the sum of the cross-sectional areas of these latter (the gas density \(\gamma\) cancels in the numerator and denominator). Such a value of the argument in the expression for the heat-transfer coefficient is extremely convenient, since instead of the variable velocity of the combustion gases, which depends on the temperature, we are dealing with a constant quantity, and thereby we are relieved of all preliminary calculations for determining the heat-transfer coefficient from the mean value of the gas velocity. This second practical reason further confirms the necessity of using the exponent \(n = 1.33\).

We proceed to determine the value of the heat-transfer coefficient in the smoke tubes, \(k_g\). The theoretical quantity of air required for the combustion of 1 kg of coal, calculated by the formula:

\[ \mathrm{L}_0=\frac{1}{23.6}\left(\frac{8}{3}\mathrm{C}'_0+8\mathrm{H}'_0+\mathrm{S}'_0-\mathrm{O}'_0\right), \]

in Syromyatnikov’s example is equal to \(\mathrm{L}_0=10.45\ \mathrm{kg/kg}\); there the excess air was also taken as \(\alpha=1.4\). Per hour, \(B'_h=847\ \mathrm{kg/h}\) of coal is burned (after deducting mechanical losses), and since \(\beta=0.59\), then \((1-\beta)=0.41\) of the total quantity of combustion gases goes into the fire tubes. Consequently, the second-by-second weight quantity of combustion gases passing through the fire tubes will be:

\[ G_g\,\sec=\frac{(\alpha L_0+1)B'_h\cdot(1-\beta)}{3600} =\frac{(1.4\cdot10.55+1)\cdot847\cdot0.41}{3600}= \]

\[ =\frac{15.63\cdot847\cdot0.41}{3600}=1.51\ \mathrm{kg/sec}, \quad\text{and since } \Omega_g=0.199\ \mathrm{m}^2, \]

\[ \text{then }\frac{G_g\,\sec}{\Omega_g}=\frac{1.51}{0.199} =7.58\ \frac{\mathrm{kg}}{\mathrm{sec}-\mathrm{m}^2}. \]

According to Syromyatnikov (p. 71),

\[ k_g=2.2+0.33\left(\frac{G_g\,\sec}{\Omega_g}\right)^{0.8} =2.2+0.33\cdot7.58^{0.8}=2.2+0.33\cdot5.06= \]

\[ =2.2+1.67=3.87\ \frac{\mathrm{Cal}}{\mathrm{m}^2-\mathrm{h}-{}^\circ\mathrm{C}^{4/3}}. \]

We now have all the data for constructing the curve \(\mathrm{H}_g=\psi(T_g)\) for the fire tubes.

We choose the scales: for \(\xi_1=-(M+2NT)\), \(w=0.01\ \mathrm{mm}\); for \(\xi_2=k(T-t)^n\), \(v=0.005\ \mathrm{mm}\); and for temperatures, \(f=0.5\ \mathrm{mm}\); then for heating surfaces we obtain the scale:

\[ 1\ \mathrm{m}^2=\frac{w\cdot f}{v}=\frac{0.01\cdot0.5}{0.005}=1\ \mathrm{mm}. \]

We mark coordinate axes on the drawing (see drawing 1). Along the abscissa axis we lay off temperatures. We calculate the values \((M_g+2N_gT)\) for \(T=0^\circ\mathrm{C}\) and \(T=1000^\circ\mathrm{C}\). At \(T=0^\circ\), \((M_g+2N_gT)=M_g=(1-\beta)M=0.41\cdot3122=1280\), or, in the chosen scale, \(1280\cdot0.01=12.8\ \mathrm{mm}\);

at \(T=1000^\circ\mathrm{C}\),

\[ (M_g+2N_gT)=0.41(3122+2\cdot0.3345\cdot1000) \]

\[ =0.41(3122+669)=0.41\cdot3791=1554, \quad\text{or in scale }1554\cdot0.01=15.54\simeq15.5\ \mathrm{mm}. \]

We lay off these quantities downward from the corresponding points of the temperature axis and draw a straight line through them.

We proceed to the construction of the curve \(\xi_2=k(T-t)^n\). To facilitate the calculations, the following Table I is used, in which are given already computed values of \((T-t)^{4/3}\) for various values of this difference at intervals of \(25^\circ\mathrm{C}\).

Table I.

In the temperature scale we have chosen it is sufficient to take points of the curve $\xi_2 = k_g (T-t)^{4/3}$ every $50^\circ$ C, within the limits from $(T-t)=100^\circ$ C to $(T-t)=800^\circ$ C. In the present case ($p_k = 13\ \text{at abs}$) the water temperature in the boiler is $t = 190^\circ$ C, i.e., for the initial temperature $T_1 = 992^\circ$ C there will be $(T-t) = (992 - 190) = 802^\circ$ C.

We compile the corresponding Table II of the values of the ordinates of the curve $\xi_2$.

Table II.

$v = 0.005.$                      $k_g = 3.87.$

The axis of abscissas, from the point corresponding to $t = 190^\circ$ C, is divided into intervals of $50^\circ$ C, i.e. of 25 mm in length, and on the corresponding verticals we lay off the values of the ordinates $\xi_2$ given in Table II. Through the points obtained we draw a smooth curve. The construction of the curve $H = \psi(T)$ has already been described above. We begin it from the point $T_1 = 992^\circ$ C. The broken line obtained is described by a curve.

From the axis of ordinates, from the point corresponding, in the scale found, to the quantity $H_g = 86.7\ \text{m}^2$ (1 mm = 1 m²), i.e. at a distance of 86.7 mm upward from the axis of abscissas, we draw a horizontal line to its intersection with the curve

\(H_g = \psi(T)\). From the point of intersection obtained, we draw a vertical line, which gives us on the abscissa axis the required temperature \(T_{3g} = 348^\circ C\)—the temperature with which the combustion gases leave the smoke tubes.

The same temperature, found analytically by trial substitutions into equation (7), was determined as \(T_{3g} = 346^\circ C\). Consequently, the accuracy of the proposed graphical method is very high.

We proceed to determine the temperature of the combustion gases at the beginning of the superheater elements. It is sufficient to carry out this construction within the limits from \(790^\circ C\) to \(990^\circ C\) (i.e., for \(T - t\) from \(600^\circ\) to \(800^\circ\)).

In view of the smallness of \(H_{ж'} = 6.4\ \mathrm{m^2}\), we choose the scales: \(w = 0.01\ \mathrm{mm}\),

\[ v = 0.001\ \mathrm{mm}\quad \text{and}\quad f = 1\ \mathrm{mm}, \]

then the scale of heating surfaces will be:

\[ 1\ \mathrm{m^2}=\frac{w\cdot f}{v}=\frac{0.01\cdot 1}{0.001}=10\ \mathrm{mm}. \]

For \(T = 790^\circ C\),

\[ (M_{ж'} + 2N_{ж'}T)=0.59(3122+2\cdot 0.3345\cdot 790) \]

\[ =0.59(3122+529)=0.59\cdot 3651=2154, \]

or, in scale,

\[ 2154\cdot 0.01=21.54 \simeq 21.5\ \mathrm{mm}; \]

for \(T = 990^\circ C\),

\[ (M_{ж'} + 2N_{ж'}T)=0.59(3122+2\cdot 0.3345\cdot 990) \]

\[ =0.59(3122+662)=0.59\cdot 3784=2233, \]

or, in scale,

\[ 2233\cdot 0.01=22.33 \simeq 22.3\ \mathrm{mm}. \]

Through the ordinates found we draw the straight line \(\xi_1\).

As regards the heat-transfer coefficient in the first part of the fire tubes, according to Syromyatnikov (see The Thermal Process of a Steam Locomotive, p. 138), its dependence on the tube diameter, all other conditions being equal, is expressed by the equation:

\[ \frac{k}{k'}=\left(\frac{d^1}{d}\right)^{0.214} \]

For our case (\(d_{ж'} = 100\ \mathrm{mm}\)) this gives:

\[ k_{ж'}=0.847\cdot k_g=0.847\cdot 3.87=3.28\ \frac{\mathrm{Cal}}{\mathrm{m^2}-\mathrm{h}-^\circ C^{4/3}} \]

The calculations of the ordinates

\[ \xi_2 = k_{ж'}(T-t)^{4/3} \]

are given in Table III.

Table III.

We lay off the values found on the corresponding verticals and, through the points obtained, draw a smooth curve (see drawing 2). By constructing, from the initial temperature \(T_1 = 992^\circ C\), the curve \(H_{ж'} = \psi(T)\). Drawing a horizontal line at a distance of 64 mm from the abscissa axis (\(H_{ж'}\) in scale is equal to 64 mm) to its intersection with \(H_{ж'}=\psi(T)\) and dropping...

drawing a perpendicular to the axis of abscissas, we find the sought temperature of the combustion gases at the ends of the superheater elements, \(T_{2ж} = 925^\circ\mathrm{C}\).

The analytical calculation gives \(T_{2ж} = 926^\circ\mathrm{C}\).

With the further flow of the combustion gases through the fire tube, their heat is used not only for evaporating water in the boiler, but also for superheating the steam in the elements of the superheater.

For convenience in the calculations, it is necessary here to divide the entire gas stream into two imaginary, separate streams—the so-called “evaporating” and “superheating” streams. It is assumed that, once divided before the beginning of the superheater elements, these streams do not mix thereafter. Although in reality this is, of course, not so, we still have no better calculation.

In this case the coefficient of division of the gases, \(\beta'\), showing what part of the total gas stream in the fire tubes gives up its heat only to superheating the steam, is found from the condition that the final temperatures (at entry into the smokebox) of both streams will be the same.

The temperature of the steam changes continuously along the entire length of the superheater elements, and therefore this variable quantity ought to be introduced into the calculation. But this complicates all the formulas and computations to such an extent that, for simplicity, it is necessary here to make a second assumption—to regard the steam temperature throughout the entire extent of the superheater elements as constant, equal to the mean between the temperature of the water in the boiler and the final temperature of the superheated steam.

For designs such as the Schmidt-system superheater, this assumption is very close to the truth and gives correct results.

The final temperature of the superheated steam, \(t_{\ddot{u}}\), and the coefficient of division of the gases in the fire tubes, \(\beta'\), are unknown and therefore must be assigned. Since (within narrow limits) the temperatures vary depending on the preliminarily adopted \(\beta'\) and \(t_{\ddot{u}}\), practically according to a straight-line law, it is sufficient to assign two values of \(\beta'\) and, for each of them, two more values of \(t_{\ddot{u}}\), so that their true magnitudes can be found by graphical interpolation. Thus, in all, four cases of heat transfer to the superheater and two cases of heat transfer to the water in the second part of the fire tubes are to be considered.

The values of \(\beta'\) and \(t_{\ddot{u}}\) that we assign must obviously be greater and smaller than their true values, so that it is not necessary to resort to less reliable extrapolation.

In our case, in the preliminary calculation when determining the heating surface of the superheater \(H_{\ddot{u}}\), \(t_{\ddot{u}} = 370^\circ\mathrm{C}\) was adopted. It is sufficient to introduce into our calculation the values

\[ t'_{\ddot{u}} = 370 + 10 = 380^\circ\mathrm{C} \quad \text{and} \quad t''_{\ddot{u}} = 370 - 40 = 330^\circ\mathrm{C} \]

(the larger downward difference from the initial temperature is taken because the final exact value of \(t_{\ddot{u}}\) always turns out to be less than the initial one).

For choosing the limiting values of \(\beta'\), it is most convenient to proceed as follows. Prof. Syromyatnikov, on p. 118 of his work “Investigation of the Working Process of a Locomotive Boiler and Superheater,” indicates that with oil firing the “thermal loading”

\[ \left(q \frac{\mathrm{Cal}}{m^2\text{—}h}\right) \]

of the superheater walls is twice as great as the same quantity for the walls of the fire tubes, other conditions being equal. The calculations made by me

calculations have shown that with coal firing this ratio is no longer 2, but only approximately

\[ \frac{q_{\text{sh}}}{q_{\text{f.t.}''}} = 1.6. \]

For the calculation it will be sufficient to set the limiting values of this ratio as \(1.6 \pm 0.3\), i.e. 1.9 and 1.3. The heating surfaces of the superheater and of the second part of the fire tubes are equal to \(H_{\text{sh}} = 65.0\ \mathrm{m^2}\) and \(H_{\text{f.t.}''} = 60.2\ \mathrm{m^2}\). Since all the heat given up by the gas stream is expressed through the heat load as \(Q = qH\), substituting the available quantities, we obtain:

\[ \text{I.}\quad \frac{Q_{\text{sh}}}{Q_{\text{f.t.}''}} = \frac{q_{\text{sh}}\cdot H_{\text{sh}}}{q_{\text{f.t.}''}\cdot H_{\text{f.t.}''}} = 1.9\,\frac{65.0}{60.2} = 2.051 \quad\text{and further} \]

\[ (1-\beta')_{\mathrm{I}} = \frac{1}{1+2.051} = \frac{1}{3.051} = 0.328, \]

\[ \text{whence}\quad \beta'_{\mathrm{I}} = 1 - 0.328 = 0.672. \]

\[ \text{II.}\quad \frac{Q_{\text{sh}}}{Q_{\text{f.t.}''}} = 1.3\,\frac{65.0}{60.2} = 1.403; \qquad (1-\beta')_{\mathrm{II}} = \frac{1}{1+1.403} = 0.416 \]

\[ \text{and}\quad \beta'_{\mathrm{II}} = 1 - 0.416 = 0.584. \]

For the calculation we adopt the rounded values:

\[ \beta'_{\mathrm{I}} = 0.675 \quad\text{and}\quad \beta'_{\mathrm{II}} = 0.575. \]

This method of finding the values of \(\beta'\) is adopted because this quantity, for different superheater designs, i.e. for different values of the ratio

\[ \frac{H_{\text{sh}}}{H_{\text{f.t.}''}}, \]

may fluctuate widely. Therefore, if \(\beta'\) is simply assigned at random, it is easy to take both of its calculated values on one side of the true value, and consequently it will be necessary to resort to undesirable extrapolation.

It is now possible to determine the final temperatures of the “evaporating” stream for both \(\beta'\). It is sufficient to carry out this construction within the limits from \(290^\circ \mathrm{C}\) to \(940^\circ \mathrm{C}\) (i.e. for \(T-t\) from \(100^\circ\) to \(750^\circ\)). We take the scales as: \(w = 0.01\ \mathrm{mm}\), \(v = 0.005\ \mathrm{mm}\), and \(f = 0.5\ \mathrm{mm}\), i.e. the scale of the heating surfaces is \(1\ \mathrm{m^2} = 1\ \mathrm{mm}\).

For \(\beta'_{\mathrm{I}} = 0.675\), at \(T = 940^\circ\),

\[ (M_{\text{f.t.}''} + 2N_{\text{f.t.}''}T) = 0.59(1-0.675)(3122 + 2\cdot 0.3345\cdot 940) = 0.59\cdot 0.325(3122+629) \]

\[ = 0.59\cdot 0.325\cdot 3751 = 720, \]

or, on the scale,

\[ 720\cdot 0.01 = 7.2\ \mathrm{mm}; \]

at \(T = 290^\circ\),

\[ (M_{\text{f.t.}''} + 2N_{\text{f.t.}''}T) = 0.59\cdot 0.325(3122 + 2\cdot 0.3345\cdot 290) \]

\[ = 0.59\cdot 0.325(3122+194) = 0.59\cdot 0.325\cdot 3316 = 637, \]

or, on the scale,

\[ 637\cdot 0.01 = 6.37 \simeq 6.4\ \mathrm{mm}. \]

For \(\beta'_{\mathrm{II}} = 0.575\), at \(T = 940^\circ\),

\[ (M_{\text{f.t.}''} + 2N_{\text{f.t.}''}T) = 0.59(1-0.575)(3122 + 2\cdot 0.3345\cdot 940) \]

\[ = 0.59\cdot 0.425\cdot 3751 = 941, \]

or, on the scale,

\[ 941\cdot 0.01 = 9.41 \simeq 9.4\ \mathrm{mm}; \]

at \(T = 290^\circ\),

\[ (M_{\text{f.t.}''} + 2N_{\text{f.t.}''}T) = 0.59\cdot 0.425\cdot 3316 = 832, \]

or, on the scale,

\[ 832\cdot 0.01 = 8.32 \simeq 8.3\ \mathrm{mm}. \]

Through the ordinates found we draw two straight lines, combining both cases on one drawing (see Drawing 3).

As for \(H_{\text{f.t.}'}\), we determine the heat-transfer coefficient according to Syromyatnikov, for \(d_{\text{f.t.}''} = 125\ \mathrm{mm}\):

\[ K_{\text{f.t.}''} = 0.805\cdot 3.87 = 3.12\, \frac{\mathrm{Cal}}{\mathrm{m^2}-\mathrm{h}-^\circ\mathrm{C}^{4/3}}. \]

The calculations of the ordinates \(\xi = K_{\mathrm{ж}''}(T-t)^{4/3}\) are given in Table IV.

Table IV.

| \(T-t\) | \multicolumn{2}{c}{\(K_{\mathrm{ж}''}(T-t)^{4/3}\)} | \(T-t\) | \multicolumn{2}{c}{\(K_{\mathrm{ж}''}(T-t)^{4/3}\)} | \(T-t\) | \multicolumn{2}{c}{\(K_{\mathrm{ж}''}(T-t)^{4/3}\)} |
|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| | Absolute | In mm | | Absolute | In mm | | Absolute | In mm |
| 100 | 1447 | 7.2 | 350 | 7700 | 38.5 | 600 | 15790 | 78.9 |
| 150 | 2488 | 12.4 | 400 | 9200 | 46.0 | 650 | 17580 | 87.9 |
| 200 | 3646 | 18.2 | 450 | 10760 | 53.8 | 700 | 19400 | 97.0 |
| 250 | 4920 | 24.6 | 500 | 12380 | 61.9 | 750 | 21260 | 106.3 |
| 300 | 6270 | 31.4 | 550 | 14060 | 70.3 | | | |

We lay off the quantities found on the corresponding verticals, and connect the points obtained by a smooth curve. Using the two straight lines
\(\xi_1 = -\left(M_{\mathrm{ж}''} + 2N_{\mathrm{ж}''}T\right)\), which we have constructed, from the initial temperature \(T_{\mathrm{ж}} = 925^\circ\mathrm{C}\), we find \(H_{\mathrm{ж}''}=\psi(T)\) for both values of \(\beta'\). Drawing a horizontal line at a distance of 60.2 mm from the axis of abscissas (on the scale \(H_{\mathrm{ж}''}=60.2\cdot 1 = 60.2\) mm) to its intersection with both curves, and dropping perpendiculars from the points found to the axis of abscissas, we find the required final temperatures of the “evaporative” stream of combustion gases:

\[ \text{for } \beta'_1 = 0.675 \qquad T^{\mathrm{I}}_{\mathrm{ш}} = 308^\circ\mathrm{C} \]

\[ \text{and for } \beta'_{\mathrm{II}} = 0.575 \qquad T^{\mathrm{II}}_{\mathrm{ш}} = 360^\circ\mathrm{C}. \]

Now, having all the temperatures, we calculate the total quantity of steam delivered by the boiler.

The quantity of heat transferred in the firebox:

\[ Q_{\mathrm{п}} = M(T_0-T_1)+N(T_0^2-T_1^2)= \]

\[ =3122(1617-992)+0.3345(1617^2-992^2)=3122\cdot625+0.3345\cdot1631400= \]

\[ =1951300+545700\ \mathrm{Cal/h}. \]

The quantity of heat transferred in the smoke tubes:

\[ Q_{\mathrm{к}}=1280(992-348)+0.1371(992^2-348^2)= \]

\[ =1280\cdot644+0.1371\cdot863000=824300+118300=942600\ \mathrm{Cal/h}. \]

The quantity of heat transferred in the first part of the fire tubes:

\[ Q_{\mathrm{ж}'}=1842(992-925)+0.1974(992^2-925^2)= \]

\[ =1842\cdot67+0.1974\cdot128500=123300+25400=148700\ \mathrm{Cal/h}. \]

The quantity of heat transferred to the water in the second part of the fire tubes:

1) for \(\beta'_{\mathrm{I}}=0.675\)

\[ Q^{\mathrm{I}}_{\mathrm{ж}''}=599(925-308)+0.0642(925^2-308^2)= \]

\[ =599\cdot617+0.0642\cdot760700=369600+48800=418400\ \mathrm{Cal/h}, \]

2) for \(\beta'_{\mathrm{II}}=0.575\)

\[ Q^{\mathrm{II}}_{\mathrm{ж}''}=783(925-360)+0.0839(925^2-360^2)= \]

\[ =783\cdot565+0.0839\cdot726000=442300+60900=503200\ \mathrm{Cal/h}. \]

Consequently, the total quantity of heat used for evaporation, at \(\beta'_1=0.675\), will be:

\[ Q^{\mathrm{I}}_{\mathrm{u}}=2497000+942600+148700+418400=4006700\ \mathrm{Cal/h}; \]

at \(\beta'_{\mathrm{II}}=0.575\), it will be:

\[ Q^{\mathrm{II}}_{\mathrm{u}}=2497000+942600+148700+503200=4091500\ \mathrm{Cal/h}. \]

In Syromyatnikov’s calculation, the heat expenditure for evaporation of the water preheated in the heater is taken as \(\lambda_{\mathrm{u}}=550\ \mathrm{Cal/kg}\), and, in addition, it is assumed that a 5% addition to the steam consumption by the engine goes for “auxiliary” needs in the form of saturated steam. Therefore the amount of steam passing through the superheater will be:

\[ B'_{\mathrm{m}}=\frac{4006700}{550\cdot 1.05}=6938\ \mathrm{kg/h} \]

and

\[ B''_{\mathrm{m}}=\frac{4091500}{550\cdot 1.05}=7085\ \mathrm{kg/h}. \]

We proceed to the determination of the final temperatures of the “superheater” gas stream, starting from the four cases assumed by us.

For determining the coefficient of heat transfer in the superheater, Prof. Syromyatnikov gives only the formula

\[ K_{\ddot{\mathrm{u}}}=21\left(\frac{3B_{\mathrm{h}}}{H_{\mathrm{zh}}+H_{\ddot{\mathrm{u}}}}\right)^{0.7} =21\left(\frac{0.59\cdot 643}{66.6+65.0}\right)^{0.7} = \]

\[ =21\left(\frac{0.59\cdot 643}{131.6}\right)^{0.7} =21\cdot 2.885^{0.7}=21\cdot 2.096=44.1\ \frac{\mathrm{Cal}}{\mathrm{m^2-h-^\circ C}} \]

(see The Thermal Process of Steam, p. 151), where \(B_{\mathrm{h}}\) is the quantity of fuel burned per hour, “reduced” to oil. This formula is given for exponent \(n=1\) and is not especially accurate, but, in the absence of a better one, we must use it.

Since \(n=1\), and moreover, according to the formula given, \(K_{\ddot{\mathrm{u}}}\) does not depend on \(T\), then \(\xi_2=K(T-t)^n\) in this case becomes a straight line intersecting the abscissa axis at \(T=t\). Owing to this, all the calculations are considerably shortened, and this is an undoubted advantage of the formula given for \(K_{\ddot{\mathrm{u}}}\).

Let us first consider the case with both \(\beta'\) values and with the assumed superheating temperature \(t''=380^\circ C\). Then the mean temperature of the steam in the superheater, entering the calculation as a fictitious constant quantity, will be:

\[ t'_{\mathrm{m}}=\frac{190+380}{2}=285^\circ C. \]

We leave the scales as before, i.e. \(w=0.01\ \mathrm{mm}\), \(v=0.005\ \mathrm{mm}\), and \(f=0.5\ \mathrm{mm}\), and for \(H_{\ddot{\mathrm{u}}}\), \(1\ \mathrm{m^2}=1\ \mathrm{mm}\).

1) \(\beta'_1=0.675\) and \(t'_{\mathrm{m}}=285^\circ C\); then at \(T=285^\circ C\)

\[ (M_{\ddot{\mathrm{u}}}+2N_{\ddot{\mathrm{u}}}T) =0.59\cdot 0.675(3122+2\cdot 0.3345\cdot 285) =0.59\cdot 0.675(3122+191) =0.59\cdot 0.675\cdot 3313=1319, \]

or, on the scale,

\[ 1319\cdot 0.01=13.19\simeq 13.2\ \mathrm{mm}; \]

at \(T=925^\circ C\)

\[ (M_{\ddot{\mathrm{u}}}+2N_{\ddot{\mathrm{u}}}T) =0.59\cdot 0.675(3122+2\cdot 0.3345\cdot 925) =0.59\cdot 0.675(3122+619) \]

\[ =0.59\cdot 0.675\cdot 3741=1490, \]

or, on the scale,

\[ 1490\cdot 0.01=14.9\ \mathrm{mm}. \]

2) \(\beta'_{\mathrm{II}} = 0.575\) and \(t'_m = 285^\circ\mathrm{C}\), then at \(T = 285^\circ\mathrm{C}\)

\[ (M_{\ddot u} + 2N_{\ddot u}T) = 0.59 \cdot 0.575 \cdot 3313 = 1124, \]

or, on the scale,

\[ 1124 \cdot 0.01 = 11.24 \simeq 11.2\ \mathrm{mm}; \]

at \(T = 925^\circ\mathrm{C}\)

\[ (M_{\ddot u} + 2N_{\ddot u}T) = 0.59 \cdot 0.575 \cdot 3741 = 1269, \]

or, on the scale,

\[ 1259 \cdot 0.01 = 12.69 = 12.7\ \mathrm{mm}. \]

In both the first and the second cases the line \(K_{\ddot u}(T-t)\) will be one and the same, since this expression does not depend on \(\beta'\).

At \(T = 285^\circ\mathrm{C}\), \(K_{\ddot u}(T-t)=0\) (it intersects the axis of abscissas), and at \(T = 925^\circ\mathrm{C}\)

\[ K_{\ddot u}(T-t) = 44.1 \cdot 640 = 28200, \]

or, on the scale,

\[ 28200 \cdot 0.005 = 141\ \mathrm{mm}. \]

As in determining the temperature of the “evaporative” stream, we combine both cases on one drawing (see drawing 4). Constructing the curves \(H_{\ddot u}=\varphi(T)\), we find that for \(H_{\ddot u}=65.0\ \mathrm{m}^2\), the final temperature of the “superheater” stream of combustion gases will be:

\[ \text{in the first case } T'_{\mathrm{п}} = 367^\circ\mathrm{C} \]

and

\[ \text{in the second case } T''_{\mathrm{п}} = 340^\circ\mathrm{C}. \]

Now let us consider the following two cases with both \(\beta'\)’s and with the assumed superheat temperature \(t''_{\ddot u}=330^\circ\mathrm{C}\).

Then

\[ t''_m=\frac{190+330}{2}=260^\circ\mathrm{C}. \]

The scales remain the same.

3) \(\beta'_{\mathrm{I}}=0.675\) and \(t''_m=260^\circ\mathrm{C}\); the values \((M_{\ddot u}+2N_{\ddot u}T)\) at \(T=285^\circ\) and \(925^\circ\) are the same as in case 1.

4) \(\beta'_{\mathrm{II}}=0.575\) and \(t''_m=260^\circ\mathrm{C}\); the values \((M_{\ddot u}+2N_{\ddot u}T)\) at \(T=285^\circ\) and \(925^\circ\) are the same as in case 2.

The line \(K_{\ddot u}(T-t)\) is one and the same for the 3rd and 4th cases. At \(T=260^\circ\mathrm{C}\)

\[ K_{\ddot u}(T-t)=0, \]

and at \(T=925^\circ\mathrm{C}\)

\[ K_{\ddot u}(T-t)=44.1(925-260)=44.1 \cdot 665 = 29300, \]

or, on the scale,

\[ 29300 \cdot 0.005 = 146.5\ \mathrm{mm}. \]

Combining these two cases also on one drawing (drawing 5), by construction we find the final temperature of the “superheater” stream:

\[ \text{in the third case } T'_{\mathrm{п}} = 346^\circ\mathrm{C} \]

and

\[ \text{in the fourth case } T''_{\mathrm{п}} = 322^\circ\mathrm{C}. \]

Now we can determine the amount of heat that went to drying and superheating the steam in all four cases:

\[ \begin{aligned} Q'_{\ddot u} &= M^I_{\ddot u}(T_{2ж}-T'_{\mathrm{п}}) +N^I_{\ddot u}(T_{2ж}^{2}-T_{\mathrm{п}}'^2) \\ &=1243(925-367)+0.1332(925^2-367^2) =1243 \cdot 558 + 0.1332 \cdot 720900 \\ &=693600+96000=789600\ \mathrm{cal/h}. \end{aligned} \]

\[ \begin{aligned} Q''_{\ddot u} &=1059(925-340)+0.1135(925^2-340^2) \\ &=1059 \cdot 585+0.1135 \cdot 740000 \\ &=619500+84000=703500\ \mathrm{cal/h}. \end{aligned} \]

\[ \begin{aligned} Q'''_{\ddot u} &=1243(925-346)+0.1332(925^2-346^2) \\ &=1243 \cdot 579+0.1332 \cdot 735900 \\ &=719700+98100=817800\ \mathrm{cal/h}. \end{aligned} \]

\[ \begin{aligned} Q''''_{\ddot u} &=1059(925-322)+0.1135(925^2-322^2) \\ &=1059 \cdot 603+0.1135 \cdot 751900 \\ &=638600+85400=724000\ \mathrm{cal/h}. \end{aligned} \]

In the first and third cases this amount of heat went to drying and superheating the amount of steam \(B^I_{\mathrm{п}} = 6938\ \mathrm{kg/h}\), i.e., the increase in heat content was:

\[ \Delta \lambda_{\ddot{u}}'=\frac{Q_{\ddot{u}}'}{B_m^I}=\frac{789600}{6938}=113.8\ \mathrm{cal/kg} \quad \text{and} \quad \Delta \lambda_{\ddot{u}}''=\frac{817800}{6938}=117.9\ \mathrm{cal/kg}. \]

Hence the heat content of the superheated steam and, from this latter, its temperature (from the Mollier diagram) are determined as:

\[ \lambda_{\ddot{u}}'=645+113.8=758.8\ \mathrm{cal/kg} \quad \text{and} \quad t_{\ddot{u}}'=357^\circ \mathrm{C} \]

and

\[ \lambda_{\ddot{u}}''=645+117.9=762.9\ \mathrm{cal/kg} \quad \text{and} \quad t_{\ddot{u}}''=365^\circ \mathrm{C}. \]

In the second and fourth cases

\[ B_m^{II}=7085\ \mathrm{kg/h}, \quad \text{i.e.} \quad \Delta \lambda_{\ddot{u}}^{II'}=\frac{703500}{7085}=99.3\ \mathrm{cal/kg} \]

and

\[ \Delta \lambda_{\ddot{u}}^{II''}=\frac{724000}{7085}=102.2\ \mathrm{cal/kg}, \]

and consequently

\[ \lambda_{\ddot{u}}^{II'}=645+99.3=744.3\ \mathrm{cal/kg} \quad \text{and} \quad t_{\ddot{u}}^{II'}=329^\circ \mathrm{C} \]

\[ \lambda_{\ddot{u}}^{II''}=645+102.2=747.2\ \mathrm{cal/kg} \quad \text{and} \quad t_{\ddot{u}}^{II''}=335^\circ \mathrm{C}. \]

If, from the found \(t_{\ddot{u}}\), we begin to determine the corresponding \(t_m\), then we shall see that they are either greater or less than the calculated \(t_m'=285^\circ\mathrm{C}\) and \(t_m''=260^\circ\mathrm{C}\). In order to find those values of \(t_{\ddot{u}}\) for which the \(t_m\) computed from them will be equal to those taken in the calculation, we compare the cases: the 1st with the 3rd and the 2nd with the 4th, and by graphical interpolation find the required \(t_{\ddot{u}}\). We do this as follows: on the axis of abscissas we lay off, as the argument, the preliminarily assumed \(t_{\ddot{u}}\); on the axis of ordinates we lay off the temperatures obtained from the calculation. It will be sufficient for us to take abscissas and ordinates from \(300^\circ\) to \(380^\circ\) (the scale is arbitrary, e.g., \(1^\circ\mathrm{C}=2\ \mathrm{mm}\)).

Let us consider the 1st and 3rd cases. When \(t_{\ddot{u}}''=330^\circ\) was assumed (\(t_m''=260^\circ\)), \(t_{\ddot{u}}''=365^\circ\) was obtained; and when \(t_{\ddot{u}}'=380^\circ\) was assumed (\(t_m'=285^\circ\)), \(t_{\ddot{u}}'=357^\circ\) was obtained. Through these points we draw a straight line expressing the dependence of the obtained \(t_{\ddot{u}}\) on the preliminarily assumed \(t_{\ddot{u}}\) (Fig. 6); we draw still another straight line passing through the origin of coordinates and bisecting the angle between them; the point of intersection of this straight line with the line for \(t_{\ddot{u}}^I\) gives the correct value of \(t_{\ddot{u}}\) for the given \(\beta'\). On the same drawing we draw a straight line expressing the dependence of the final temperature of the “superheater” flow on the preliminarily assumed \(t_{\ddot{u}}\). To construct it we have that, with the assumed \(t_{\ddot{u}}''=330^\circ\), \(T_n''=346^\circ\) was obtained, and with the assumed \(t_{\ddot{u}}'=380^\circ\), \(T_n'=367^\circ\) was obtained. Through these points we draw a straight line and, dropping from the just-found point for \(t_{\ddot{u}}^I\) a vertical to its intersection with this line, we find the corresponding value of \(T_n^I\). There is no need to measure on the ordinate axis the numerical values of \(t_{\ddot{u}}^I\) and \(T_n^I\), since these are only intermediate, auxiliary quantities.

In exactly the same way we compare the 2nd and 4th cases (\(\beta_{II}'=0.575\)). We have the points: with the assumed \(t_{\ddot{u}}''=330^\circ\), \(t_{\ddot{u}}^{II''}=335^\circ\) and \(T_n^{II''}=322^\circ\) were obtained; and with the assumed \(t_{\ddot{u}}'=380^\circ\), \(t_{\ddot{u}}^{II'}=329^\circ\) and \(T_n^{II'}=340^\circ\) were obtained.

Drawing straight lines through these points (see Fig. 7), at the intersection of the line for \(t_{\ddot{u}}^{II}\) with the straight line inclined at an angle of \(45^\circ\), we find the correct-

...the value \(t_{\mathrm{i}}^{II}\) for \(\beta'_{II}=0.575\), and, dropping a vertical from the point found to the intersection with the line for \(T_{\mathrm{p}}^{II}\), we find the corresponding value \(T_{\mathrm{p}}^{II}\).

Now it is possible to determine the exact value of \(\beta'\), and then the final values of \(t_{\mathrm{i}}\) and \(T_{\mathrm{ezh}}\). To do this, along the abscissa axis we lay off the values \(\beta'\) (in our case from 0.575 to 0.675), and along the ordinate axis the temperatures (as in the preceding two diagrams, we take them from \(300^\circ\) to \(380^\circ\)).

We place Figs. 6, 7, and 8 side by side in such a way that their abscissa axes lie on one straight line, with Fig. 8 in the middle, Fig. 6 on the right, and Fig. 7 on the left.

Having arranged the diagrams in this way, we transfer horizontally the values \(t_{\mathrm{i}}\) and \(T_{\mathrm{p}}\) obtained in Figs. 6 and 7 to the verticals corresponding to both \(\beta'\). We join the points obtained by straight lines: \(t_{\mathrm{i}}=f_1(\beta')\) and \(T_{\mathrm{p}}=f_2(\beta')\). In addition, we draw the straight line \(T_{\mathrm{p}}=f_3(\beta')\), for the construction of which we have, at \(\beta'_{II}=0.575\), \(T_{\mathrm{p}}^{II}=360^\circ\mathrm{C}\), and at \(\beta'_I=0.675\), \(T_{\mathrm{p}}^{II}=308^\circ\mathrm{C}\).

Dropping from the point of intersection of the straight lines \(T_{\mathrm{p}}=f_2(\beta')\) and \(T_{\mathrm{p}}=f_3(\beta')\) a perpendicular to the abscissa axis, we find the sought value of \(\beta'\), at which the condition is satisfied that the final temperatures of the “evaporative” and “superheater” streams must be equal. At the point where this same vertical intersects the line \(t_{\mathrm{i}}=f_1(\beta')\), we obtain the corresponding value of \(t_{\mathrm{i}}\).

In Fig. 8 we obtained: \(\beta'=0.617\), \(T_{\mathrm{ezh}}=T_{\mathrm{p}}=T_{II}=338^\circ\mathrm{C}\), and \(t_{\mathrm{i}}=345^\circ\mathrm{C}\).

The exact analytical calculation carried out gave, for the same quantities, the values: \(\beta'=0.618\), \(T_{\mathrm{ezh}}=336^\circ\mathrm{C}\), and \(t_{\mathrm{i}}=346^\circ\mathrm{C}\). From this it is evident that the accuracy of the proposed graphical method is very high.

It should also be noted here that a more precise determination of the values of the heat-transfer coefficients is necessary, especially in the superheater; this can best be done by setting up special laboratory experiments.

Drawing 1

Compiled by K. P. Schmidt.

Visible annotations:

• \(h_g(f-t)^{1/3}\)
• \(h_g(f_0-t)^{1/3}\)
• \(R_y\)
• \(190^\circ\)
• \(245^\circ\)
• \(398^\circ\)
• \(100^\circ\)
• \((\zeta_y+\xi_n)\ [[unclear]]\)
• \((\zeta_y+\xi_n)= [[unclear]]\)
• \((\xi_y+\xi_n)\ [[unclear]]\)

Scales:

• \(W = 0.005\)
• \([[unclear]] = 0.0025\)
• \(\beta = 0.05\)
• \(t_m = 0.5\ \mathrm{mm}\)

Drawing 7

Visible annotations:

• \(330^\circ\)
• \(380^\circ\)
• \(t_m\)
• \(S_m\)

Drawing 8

Visible annotations:

• \(\beta'_x = 0.617\)
• \(\beta'_x\)
• \(0.530\)
• \(0.540\)
• \(0.600\)
• \(0.610\)
• \(0.620\)
• \(0.630\)
• \(0.640\)
• \(0.650\)
• \(0.660\)
• \(0.670\)
• \(t_m = 345^\circ\)
• \(S_{m} = 338^\circ\)
• \(t_m\)
• \(S_m\)
• \(330^\circ\)
• \(340^\circ\)
• \(350^\circ\)
• \(360^\circ\)
• \(370^\circ\)
• \(380^\circ\)

Drawing 6

Visible annotations:

• \(330^\circ\)
• \(340^\circ\)
• \(t_m\)
• \(S_m\)

For the article by K. P. Schmidt.

Diagram 2

Visible labels and annotations:

• Vertical axis: \(\mathfrak{H}_{ж}\)
• Horizontal axis: \(\vartheta\)

Scales:

\[ W = 0.01 \]

\[ v = 0.001 \]

\[ f = 1 \]

\[ 1\ \mathrm{m}^2 = 10\ \mathrm{mm} \]

Curve and construction labels:

\[ \xi_2 = k_{ж}(\vartheta - t)^{4/3} \]

\[ \mathfrak{H}_{ж} - k(\vartheta) \]

\[ \xi_1 = -\left(M'_{ж} + 2NF_{ж}\vartheta\right) \]

Numerical markings visible on the construction include:

\[ 815,\quad 925^\circ,\quad 996^\circ \]

Diagram 5

Visible labels and annotations:

• Vertical axis: \(\mathfrak{H}_{i}\)
• Horizontal axis: \(\vartheta\)

Curve label:

\[ k_{з}(\vartheta - t) \]

Numerical markings visible on the construction include:

\[ 260^\circ,\quad 392^\circ,\quad 346^\circ,\quad 285^\circ,\quad 925^\circ \]

Parameter labels at the lower right:

\[ \beta''_{i} = 0.575 \]

\[ \beta'_{i} = 0.575 \]

To the article by K. P. Schmidt.

Fig. 3

\(H_{ж}^{*}\)

Scales:

\(w = 0.01\)
\(v = 0.0025\)
\(f = 0.25\)

\(1\ \mathrm{m}^{2} = 1\ \mathrm{mm}\)

\(J\)

\(32^\circ\)
\(36^\circ\)
\(93^\circ\)
\(94^\circ\)

\(\beta_{ж}^{\prime} = 0.575\)
\(\beta_{ж}^{\prime\prime} = 0.575\)

Fig. 4

\(H_{i}^{\circ}\)

\(k\mu(5 - t)\)

Scales:

\(w = 0.01\)
\(v = 0.0025\)
\(f = 0.25\)

\(1\ \mathrm{m}^{2} = 1\ \mathrm{mm}\)

\(J\)

\(34^\circ\)
\(36^\circ\)
\(28.5^\circ\)
\(98^\circ\)

\(\beta_{i}^{\prime} = 0.575\)
\(\beta_{i}^{\prime\prime} = 0.575\)