Stud. K. P. Schmidt.

A New Graphical Method for Calculating Heat Transfer in a Steam Locomotive Boiler

In solving problems of heat transfer through a wall from one medium to another, the starting point is the following two differential equations

\[ \frac{dQ}{\tau}=\kappa (T-t)^n\,dH \ldots \ldots \ldots \ldots (1) \]

and

\[ \frac{dQ}{\tau}=-(M+2NT)\,dT \ldots \ldots \ldots \ldots (2) \]

where:

\(dQ\) is the quantity of heat transferred in the time interval \(\tau\) from one medium to another; it is negative for the heat source and positive for the body being heated; \(\kappa\) is the heat-transfer coefficient; \(T\) is the temperature of the heat source and \(t\) is the temperature of the heated body, on the two sides of the surface element of the heating wall separating them, \(dH\); \((M+2NT)\) is the actual (not mean) heat capacity, at constant pressure and temperature \(T\), of the entire quantity of the heat-releasing body that washes, per unit time, the heating surface \(H\).

As for the exponent \(n\) in the term \((T-t)\), different investigators assign it different values. According to Redtenbacher \(n=1\); according to Rankine \(n=2\); and according to the investigations of Prof. Syromyatnikov, the most reliable value of this exponent is \(n=\frac{4}{3}\) (see Syromyatnikov, Investigation of the Working Process of a Steam Locomotive Boiler and Superheater). It should be noted here that Prof. Syromyatnikov’s works are based on the very extensive experimental material of Prof. Lomonosov’s tests on steam locomotives, and therefore deserve greater confidence.

Returning to equations (1) and (2), we see that their left-hand sides are equal, and consequently the right-hand sides are equal as well, i.e.

\[ \kappa (T-t)^n\,dH=-(M+2NT)\,dT \ldots \ldots \ldots (3) \]

or

\[ \kappa dH=-\frac{M+2NT}{(T-t)^n}\,dT \ldots \ldots \ldots (4) \]

To obtain the relation between the temperature of the body giving off heat (i.e. the temperature of the combustion gases in the steam locomotive boiler) and the heating surface washed by it, it is necessary to integrate one of these differential equations, (3) or (4).

In order to obtain expressions with which it is still possible to operate in calculations, during integration one has to assume that the heat-transfer coefficient \(\kappa=\mathrm{const}\) and does not depend on \(T\). Integrating equation (4) at \(t=\mathrm{const}\) (heat transfer to the water in the boiler), from the initial temperature of the combustion gases \(T_0\) to the temperature \(T_x\) with which they leave, after washing the heating surface \(H\), we obtain:

1) for \(n=1\) (Redtenbacher’s hypothesis)

\[ \varkappa' H=(M+2Nt)\ln\frac{T_0-t}{T_x-t}+2N(T_0-T_x)\ldots \tag{5} \]

(the integration process, see Syromyatnikov, The Thermal Process of the Steam Locomotive, p. 52).

2) for \(n=2\) (Rankine’s hypothesis)

\[ \varkappa'' H=(M+2Nt)\frac{T_0-T_x}{(T_x-t)(T_0-t)} +2N\ln\frac{T_0-t}{T_x-t}\ldots \tag{6} \]

(integration, see ibid., p. 68).

and 3) for \(n=\frac{4}{3}\) (Syromyatnikov’s work)

\[ \varkappa''' H= 3(M+2Nt)\left[ \frac{1}{\sqrt[3]{T_x-t}}-\frac{1}{\sqrt[3]{T_0-t}} \right] + \]

\[ +3N\left[ \sqrt[3]{(T_0-t)^2}-\sqrt[3]{(T_x-t)^2} \right]\ldots \tag{7} \]

If, by one of these equations, it is necessary to find the required heating surface from a prescribed temperature \(T_x\), this presents no particular difficulty. But in the majority of cases it is necessary, for a known heating surface—\(H\), to find the temperature of the combustion gases \(T_x\) which they will have after washing this surface. For such calculations equations (5), (6), and (7) are very inconvenient, since they cannot be solved algebraically. Here one can proceed only by trial substitutions, and this is rather tedious work.

It is true that these equations, by algebraic transformations, can be given a somewhat more convenient form, for example:

\[ \text{1) for } n=1 \quad \lg(T_x-t)=a_1-b_1T_x\ldots \tag{8} \]

\[ \text{2) for } n=2 \quad (T_x-t)\lg(T_x-t)=a_2+b_2(T_x-t)\ldots \tag{9} \]

\[ \text{and 3) for } n=\frac{4}{3}\quad (T_x-t)=a_3+b_3\sqrt[3]{T_x-t}\ldots \tag{10} \]

where

\[ a_1=\lg(T_0-t)+\frac{2NT_0-\varkappa'H}{2.3026(M+2Nt)}\ldots \tag{11} \]

\[ b_1=\frac{2N}{2.3026(M+2Nt)}\ldots \tag{12} \]

\[ a_2=\frac{M+2Nt}{2.3026}\ldots \tag{13} \]

\[ b_2=2N\lg(T_0-t)- \frac{M+2Nt+\varkappa''H(T_0-t)} {2.3026(T_0-t)}\ldots \tag{14} \]

\[ a_3=\frac{M+2Nt}{N}\ldots \tag{15} \]

and

\[ b_3=\sqrt[3]{(T_0-t)^2} -\frac{3M+6Nt+\varkappa'''H\sqrt[3]{T_0-t}} {3N\sqrt[3]{T_0-t}}\ldots \tag{16} \]

Equations (8), (9), and (10), after substituting into them the values of the coefficients calculated by formulas (11)—(16), are more convenient for calculation. They may be solved either analytically, by trial substitutions, or graphically, by constructing curves of the functions of \((T_x - t)\) that separately represent the right- and left-hand sides of the equations; from the intersection of two such curves we obtain, on the \(T_x\) axis, the value of the root of the equation.

Besides this way of determining heat transfer in a steam-locomotive boiler, there is also Prof. Grinevetsky’s graphical method, set forth in detail by Prof. Syromyatnikov (The Thermal Process of a Steam Locomotive, p. 55). This method is suitable only when \(n = 1\) and \(k = \mathrm{const}\), i.e., for Redtenbacher’s hypothesis and a constant heat-transfer coefficient.

Meanwhile, the very thorough work of Prof. Syromyatnikov has shown that, for example, in the fire tubes of a steam-locomotive boiler the heat-transfer coefficient varies according to the law:

\[ K_g = 6 + 2{,}45\, W_g^{0{,}7} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (17) \]

where \(W_g\) is the velocity (in m/sec) of the flow of combustion gases in the tubes. This velocity can always be expressed through the per-second weight quantity of combustion gases passing through the fire tubes, through their temperature, and through the sum of the cross-sectional areas of all tubes, i.e., ultimately \(k = f(T)\).

But an attempt to introduce a variable \(k\) into equation (4), after integration, gives incredibly complicated expressions, already quite unsuitable for practical work with them. Grinevetsky’s method does not at all allow for a variable heat-transfer coefficient.

Meanwhile, the conventional adoption of a constant \(k\) forces one, in calculations, in order to determine it, to assume some value of \(T_x\), to determine from it the average velocity of gas flow in the tubes, and from this to find the average value of \(k\). If, after substituting this quantity into the heat-transfer equation, a value of \(T_x\) is obtained that differs noticeably from that assumed in determining \((W_g)_{\mathrm{avg}}\) and \(k_{\mathrm{avg}}\), then the whole procedure is begun again, until the preliminary value of \(T_x\) has been chosen sufficiently successfully.

The average value of the heat-transfer coefficient differs from its actual value at the ends of the fire tubes (i.e., at \(T_o\) and \(T_x\)) by 20—30%. This deviation causes an incorrect outline of the calculated temperature curve along the length of the fire tubes (or flame tubes—it is immaterial), i.e., the curve \(T = f(H)\). The actual temperature curve has a large curvature, as is seen in Fig. 1. Adoption of Redtenbacher’s hypothesis, i.e., \(n = 1\), further increases this deviation.

This inaccuracy is insignificant if the calculation concerns a steam locomotive with fire tubes about 4.5 mtr long (this was the length of almost all locomotives on which Lomonosov carried out his experiments), and if at the same time we are not interested in the outline of the curve \(T = f(H)\). But already, for example, a transition to longer tubes may cause a noticeable error in determining \(T_x\), and where it is important to know the outline of \(T = f(H)\), as, for example, in determining heat transfer in a superheater, there the difference becomes quite appreciable. This inaccuracy

and the necessity of carrying out preliminary trial calculations in order to determine \(k\) indicates the imperfection of the accepted methods of heat calculation.

All these shortcomings can be avoided if equation (3) is not integrated in its general form, but is represented as:

\[ \frac{dH}{dT}=-\frac{M+2NT}{k(T-t)^n}\ . . . . . . . . . . . (18) \]

compute this derivative of \(H\) with respect to \(T\) for the given particular case, and perform the integration graphically in order to obtain \(H=f(T)\).

Here two methods are possible. The first is, for any \(k\) and \(n\), to compute the values of the right-hand side of equation (18) for several values of \(T\), beginning with \(T_0\), and from the points found to construct a smooth curve

\[ \frac{dH}{dT}=\varphi(T) \]

and, by planimetering the area between it, the \(T\)-axis, and the boundary ordinates of the temperatures \(T_0\) and \(T_x\), to find \(H=\psi(T)\), as is shown in Fig. 2.

For finding \(T_x\) from a given \(H\), this method is inconvenient, since it requires determining it by trial planimetering until an area equal to the given \(H\) is obtained.

The second method consists in the graphical determination of the magnitude of the ratio of the numerator and denominator of the right-hand side of equation (18), and in the graphical obtaining from it of the curve \(H=\psi(T)\). This construction is carried out as indicated in Fig. 3.

Below the \(T\)-axis a straight line is drawn—\((M+2NT)=\xi_1(T)\); above it, from computed points, the curve \(k(T-t)^n=\xi_2(T)\) is constructed. Then, as shown in the drawing, starting from \(T_0\), we divide both these curves by a series of vertical lines into segments: \(aa_1\), \(bb_1\), \(cc_1\), etc. (the segments may be of any size and may be unequal). Each of these sections is divided in half by verticals: \(1'n_1\), \(2'n_2\), etc. With the aid of compasses, or a \(45^\circ\) set square, we lay off the magnitude \(\xi_2(T)\) on the ordinate \(1—1'\) to the left of point 1; we obtain point \(m_1\), and join it by a straight line to point \(n_1\). From point \(p_1\), parallel to \(m_1n_1\), we draw the straight line \(p_1p_2\), up to the line \(bb_1\), i.e., to the boundary of the next section. We perform the same operation on the second section as well, drawing on it, from point \(p_2\), the line \(p_2p_3\), parallel to \(m_2n_2\), etc. The broken line thus obtained, \(p_1p_2p_3\ldots\), is inscribed in the curve \(H=\psi(T)\). That this is so is easily proved.

Indeed: the segment \(1m_1\) represents, on a certain scale, the quantity \(k(T-t)^n\), for the given \(T\) corresponding to point 1, while the segment \(1n_1\), likewise on its own scale, represents the quantity \(-(M+2NT)\), at the same \(T\). The ratio of these two quantities, i.e.

\[ -\frac{M+2NT}{k(T-t)^n}=\frac{dH}{dT}, \]

is measured in the drawing by the tangent of the angle \(\alpha\). But, as is known, the derivative of a function at a given point is equal to the tangent of the angle which the tangent to the curve (expressing the given function) at that point makes with the axis of abscissas; consequently, by drawing from the initial point \(p_1\) (corresponding to \(T_0\)) the line \(p_1p_2\), parallel to \(m_1n_1\), we obtain, approximately, a segment of the curve \(H=\psi(T)\) on the first section. The greater the number of sections we take, the more closely the resulting broken line will be inscribed in the true curve \(H=\psi(T)\).

The scale for the ordinates \(H\) is obtained from the scales adopted for constructing the functions \(\xi_1\) and \(\xi_2\), and from the scale for \(T\).

Namely, if for \(\xi_1 = -(M + 2NT)\) we choose the unit scale

\[ 1\,\frac{\mathrm{Cal}}{\mathrm{h}-{}^\circ \mathrm{C}} = w \ \mathrm{mm}, \]

for

\[ \xi_2 = k(T-t)^n \]

the unit scale

\[ 1\,\frac{\mathrm{Cal}}{\mathrm{h}-\mathrm{m}^2} = v \ \mathrm{mm}, \]

and for temperature the unit scale

\[ 1^\circ \mathrm{C} = f \ \mathrm{mm}, \]

then the scale for the heating surfaces will be

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

From the curve found, \(H = \psi(T)\), it is very simple to find, for a given \(H\), \(T_x\), and conversely—see Fig. 4.

The method of construction set forth is very flexible; it is entirely not “afraid” of a variable \(k\) and of different values of \(n\), and it requires no preliminary calculations, since it operates not with an average value of \(k\), which would have to be assumed, but with the value of the heat-transfer coefficient at the given point.

It should also be noted here that the magnitude of the exponent \(n\) characterizes the share of heat transfer by radiation in the overall heat transfer; consequently, it is possible that it is not only a fractional quantity, but also a variable one, depending on \(T\).

If the corresponding investigations show the existence of such a dependence, then for this case as well—the variable \(n\)—the method presented remains just as simple and convenient, and the most perfect and universal.

To the article by student N. P. Schmidt.

Fig. 1.

Calculated
Actual

\(y=f(t)\)
\(y_{\text{calculated}}=\) [[unclear: formula]]
\((M/d+s)\)

Fig. 2.

\(y=f(t)\)
\(\xi_{0}\)

Fig. 3.

\(y(t-t_{0})=\xi\varphi(y)\)

\((c)_{\xi}=(S,NZ+M)\)

\((-S,NZ+M)\)

Fig. 4.

\(\xi_{0}\)
\(x\)
\(x\)
\(x\)

Fig. 5.

\(\varphi(y)\)
\(\xi=f(y)=\dfrac{\xi_{0}}{k}\)