Universal Heating-Cooling Diagram of Solids in Similarity Criteria (Fo-Bi Diagrams)
Abstract Generated abstract
This paper develops a nomographic method for calculating transient heating and cooling of homogeneous solid bodies using Fourier and Biot similarity criteria. Building on standard nonstationary heat conduction theory, it derives dimensionless relations for the degree of heating uniformity and temperature potential for plates and cylinders, including cases where external exchange is governed by surface conductance, heat transfer, or their combined coefficient. The resulting Fo, Bi diagram and accompanying tables are intended to replace repeated cumbersome analytical calculations while retaining their theoretical basis. An application to furnace and coking calculations illustrates how heating time can be estimated as a function of charge thickness and how trial solutions may be used to select practical operating conditions.
Full Text
A Universal Diagram for Heating–Cooling of Solids in Similarity Criteria
(Fo—Bi Diagram)
N. N. Norkin
INTRODUCTION
The physico-mathematical essence of problems concerning the heating of bodies has been set forth with exhaustive completeness by representatives of the theory of the nonstationary temperature field in heated bodies (Gröber, Schack, M. Adams, and others). The purpose of the present work is, on the basis of this theory in combination with similarity theory, to provide a nomographic method for calculating the heating–cooling of homogeneous solid bodies by means of dimensionless complexes—similarity criteria. This is necessary because, in solving practical problems, one does not confine oneself (and cannot confine oneself) to a single solution, but carries out several trial solutions in order to adopt optimal conditions. The analytical solution of the heating–cooling of bodies, cumbersome in volume even for one particular case, when many repeated (necessary) calculations are involved, “frightens off,” “inspires distrust” toward exact methods of calculation, and forces recourse to particular (recipe-like) methods based on empirical formulas. A diagram for calculating the heating–cooling of bodies, while preserving the accuracy of physico-mathematical analysis, is free of recipe-like character and makes it possible quickly to choose the most suitable conditions for any particular case.
The author foresees that the user of the diagram will benefit if the physico-mathematical foundations of the theory of the nonstationary temperature field and the foundations of the theory of heating–cooling of bodies have been studied; but if, instead of studying the theory, one limits oneself merely to becoming acquainted with the method of calculation by the diagram, then this will bring not benefit but harm: it will produce a mechanical view of the physical essence of the complex phenomenon of heating–cooling. The author uses the word “through-heating” not as a substitute for the word “heating,” but in order to emphasize that, in the process of heating (cooling) bodies, there will always be a temperature difference at two points of these bodies (in particular, at the center and at the surface), and the magnitude of the difference permissible from the technological point of view depends on the heating time, the physical properties of the body being heated, and the influence of external conditions (heat-transfer conditions). In the same sense, the concepts of degree of uniformity of through-heating and temperature potential have been introduced into the text.
PHYSICO-MATHEMATICAL THEORY OF THE DIAGRAM
The process of heating–cooling depends on internal and external physical conditions. The internal conditions are characterized by the physico-thermal parameters of the body; the external conditions, by the method of heat transfer.
The thermophysical parameters of a body are:
thermal conductivity of the body \(\lambda\) \((\mathrm{kcal}/\mathrm{m}^{2}\,\mathrm{h}\,^\circ\mathrm{C})\);
heat capacity of the body \(c\) \((\mathrm{kcal}/\mathrm{kg}\,^\circ\mathrm{C})\);
specific weight \(\gamma\) \((\mathrm{kg}/\mathrm{m}^{3})\).
These parameters, combined in the form
\[ a=\frac{\lambda}{c\gamma}\;(\mathrm{m}^{2}/\mathrm{h}), \tag{1} \]
give a more generalized parameter—the thermal diffusivity \(a\) \((\mathrm{m}^{2}/\mathrm{h})\). Thermal diffusivity characterizes the rate of equalization of temperature in a nonuniformly heated body, or, in other words, the rate of propagation of temperature.
From the standpoint of internal conditions, the change in the temperature field in a heated–cooling body is affected, in addition to thermal diffusivity, by the geometrical dimensions of the body and the time of heating–cooling.
Consequently, the conditions are determined by Fourier’s equation
\[ \frac{\partial t}{\partial \tau} = a\left( \frac{\partial^{2}t}{\partial x^{2}} + \frac{\partial^{2}t}{\partial y^{2}} + \frac{\partial^{2}t}{\partial z^{2}} \right) \tag{2} \]
and, for physically similar (or physically identical but geometrically different) systems, by the Fourier criterion of homochronicity
\[ F_{0}=\frac{a}{x^{2}}\tau . \]
The external conditions are determined by the method of heat transfer, i.e., by the heat-exchange equation
\[ \alpha \Delta t=\lambda \frac{\partial t}{\partial x}\quad x\to 0 \tag{3} \]
and, for physically similar systems, by the Biot criterion
\[ Bi=\frac{\alpha}{\lambda}x; \tag{4} \]
here \(\alpha\) is the coefficient of heat transfer from the external medium (or to the external medium),
\(\lambda\) and \(x\) are, as above, the coefficient of thermal conductivity and the thickness of the heated (cooling) body.
A feature of muffle heating (for example, in coke ovens) is that the thermal interaction of the heated body with the heating medium is effected not directly, but through the separating wall, through the heating surface; therefore the external conditions of heat exchange are determined not only by the heat-transfer coefficient from the heating gases \((\alpha\ \mathrm{kcal}/\mathrm{m}^{2}\,\mathrm{h}\,^\circ\mathrm{C})\), but to a greater extent by the conductance of the wall \(\left(\beta=\frac{\lambda_{\mathrm{st}}}{S_{\mathrm{st}}}\ \mathrm{kcal}/\mathrm{m}^{2}\,\mathrm{h}\,^\circ\mathrm{C}\right)\).
Thus, in order to reveal the influence of external conditions on the heating–cooling process during heating–cooling through a wall, the problem may be treated in the following variants.
First variant. The external heat-exchange conditions are determined by the temperature of the heating–cooling surface (wall) on the side of the heat carrier (heat-absorbing medium). The influence of the conductance \((\beta)\) and the wall temperature \((t_{\mathrm{st}})\) on the heating–cooling process is analyzed. Re-
Fo–Bi DIAGRAM
for calculating the heating of homogeneous bodies
Degree of uniformity of heating
for a semi-infinite body
\[ \varphi=\frac{t-t_{0}}{t_{1}-t_{0}}=f(Fo,Bi) \]
\[ Fo=\frac{a\tau}{s^{2}} \]
Degree of uniformity of heating
for a plate
\[ \varphi=\frac{t-t_{0}}{t_{1}-t_{0}}=f(Fo,Bi) \]
\[ Fo=\frac{a\tau}{s^{2}} \]
Temperature potential
\[ \frac{t_{\mathrm{c}}}{t_{\mathrm{surf}}}=\psi=\varphi(1-\varphi) \]
for a plane body under double-sided heating
Curves marked:
\[ Bi=0.2,\quad Bi=0.4,\quad Bi=0.6,\quad Bi=0.8,\quad Bi=1.0,\quad Bi=2,\quad Bi=10 \]
Degree of uniformity of heating
for a semi-infinite body
\[ \varphi=\frac{t-t_{0}}{t_{1}-t_{0}}=f(Fo,Bi) \]
\[ Fo=\frac{a\tau}{s^{2}} \]
Degree of uniformity of heating
for a cylinder
\[ \varphi=\frac{t-t_{0}}{t_{1}-t_{0}}=f(Fo,Bi) \]
\[ Fo=\frac{a\tau}{s^{2}} \]
Temperature potential
\[ \frac{t_{\mathrm{c}}}{t_{\mathrm{surf}}}=\psi=\varphi(1-\varphi) \]
for a cylindrically shaped body
Curves marked:
\[ Bi=0.2,\quad Bi=0.4,\quad Bi=0.6,\quad Bi=0.8,\quad Bi=1.0,\quad Bi=2,\quad Bi=10 \]
the solution of the problem is reduced to finding the heating–cooling time in the form of a functional relation
\[ \tau=f(\beta,t_{cm},x) \tag{5} \]
or in the form of an equation in similarity criteria
\[ \tau=f_1(t_{cm},F_o,Bi), \tag{6} \]
where the criterion is
\[ Bi=\frac{\beta}{\lambda}x. \tag{7} \]
Second variant. The external conditions of heat exchange are determined by the temperature of the heat-transfer medium (heat-absorbing medium). The influence of the heat-transfer coefficient \((\alpha)\) and of the temperature of the heat-transfer medium (heat-absorbing medium) on the heating–cooling process is analyzed. The solution of the problem is reduced to finding the heating–cooling time. The solution of the problem is reduced to finding the heating–cooling time in the form of a functional relation
\[ \tau=f_2(\alpha,\beta,t_{\text{gas}},x) \tag{8} \]
or, combining \(\alpha\) and \(\beta\) into a single heat-transfer coefficient,
\[ K=\frac{1}{\frac{1}{\alpha}+\frac{1}{\beta}}\ \text{kcal}/m^2\ \text{h}^{\circ}C, \]
in the form of an equation in similarity criteria
\[ \tau=f_2(t_{\text{gas}},F_o,Bi''), \tag{9} \]
where the criterion is
\[ B''i=\frac{K}{\lambda}x. \tag{10} \]
The combined influence of the internal and external conditions on the change of the temperature field in the heated–cooled body is mathematically expressed by the equation (according to Gröber)
\[ t=t_b+(t_a-t_b)\sum_{k=1}^{k=\infty} 2\,\frac{\sin\delta_k}{\delta_k+\sin\delta_k\cos\delta_k}\, e^{-\delta_k^2\frac{a\tau}{x^2}} \cos\left(\delta_k\frac{X}{x}\right). \tag{11} \]
In this equation \(t_a\) is the temperature of the body at the beginning of heating (cooling); \(t_b\) is the temperature of the heating (cooling) medium; \(\delta_k\) are the roots of the transcendental equation, as functions of the Biot criterion, expressed either through the conductivity coefficient or through the incomplete heat-transfer coefficient, i.e.
\[ \delta_k=f\left(\frac{\alpha}{\lambda}x\right)-f\left(\frac{\beta}{\lambda}x\right)\div f\left(\frac{K}{\lambda}x\right)=f(Bi); \]
here \(\dfrac{a}{x^2}\tau\) is the Fourier criterion; \(x\) is half the thickness (for a flat plate) or the radius (for a cylindrical body); \(X\) is the current coordinate; \(a\) is thermal diffusivity; \(\alpha\) is the heat-transfer coefficient \((\text{kcal}/m^2\ \text{h}^{\circ}C)\); \(\tau\) is the heating (cooling) time in hours.
For the subsequent derivations we shall consider the origin of coordinates to be coincident with the axis of symmetry of the body under consideration. Then, to determine
temperature on the surface of the heated (cooling) body the current coordinate is \(X=x\), and equation (11) takes the form
\[ t_0=t_b+(t_a-t_b)\sum_{k=1}^{k=\infty}2\, \frac{\sin\delta_k}{\delta_k+\sin\delta_k\cos\delta_k}\, e^{-\delta_k^2\frac{a\tau}{x^2}}\cos\delta_k . \tag{12} \]
To determine the temperature on the axis (at the center) of the heated (cooling) body, the current coordinate is \(X=0\), and equation (11) takes the form
\[ t_m=t_b+(t_a-t_b)\sum_{k=1}^{k=\infty}2\, \frac{\sin\delta_k}{\delta_k\sin\delta_k\cos\delta_k}\, e^{-\delta_k^2\frac{a\tau}{x^2}} . \tag{13} \]
Analyzing equations (12) and (13), we establish that the character of the temperature distribution in the heated (cooling) body depends on the magnitudes of the determining dimensionless complexes—the Biot and Fourier criteria; moreover, as indicated, the criterion
\[ Bi=\frac{\alpha}{\lambda}x \]
determines the external conditions of heating (cooling), since its magnitude depends on the heat-transfer coefficient \((\alpha)\), or on the conductivity of the heating surface \((\beta)\), or on the heat-transfer coefficient \((K)\); the Fourier criterion
\[ Fo=\frac{a}{x^2}\tau \]
determines the internal conditions of heating (cooling), since its magnitude depends on the thermal diffusivity of the body \((a)\).
Therefore we rewrite equation (12) in the form
\[ t_0=t_b+(t_a-t_b)\Phi_0(Bi,Fo) \tag{14} \]
and equation (13) in the form
\[ t_m=t_b+(t_a-t_b)\Phi_m(Bi,Fo). \tag{15} \]
III. COORDINATES OF POINTS OF CURVES AND CONSTRUCTION OF THE DIAGRAM
Let us call the degree of uniformity of heating the ratio of the temperature on the axis (at the center) of the body to the temperature on the surface of the body after the lapse of a definite time interval:
\[ \varphi=\frac{t_m}{t_0}; \tag{16} \]
in the ideal case \(\varphi=1\), i.e. \(t_m=t_0\): heating—cooling is perfect; practically, \(\varphi=1\) is unattainable.
Having in view equations (14) and (15), and taking the initial temperature of the heated body \(t_a=0\) or of the cooling medium \(t_b=0\), which is practically quite acceptable, we obtain
\[ \varphi=\frac{1-\Phi_m(Bi,Fo)}{1-\Phi_0(Bi,Fo)}; \tag{17} \]
equation (17) makes it possible to determine the degree of uniformity of heating as a function of the two determining criteria, Fourier and Biot. With the aid of this equation, the following typical problems can be solved.
A. Known: the heat-transfer coefficients \((\alpha)\) and the duration of heating–cooling \((\tau)\). The degree of uniformity of heating–cooling \((\varphi)\) is determined. B. Given: the degree of uniformity of heating \((\varphi)\) and the heat-transfer coefficient \((\alpha)\). The time \((\tau)\) required to achieve the specified uniformity of heating–cooling is determined. C. Given: the degree of uniform-
of the uniformity of heating–cooling \((\varphi)\) and the duration of heating \((\tau)\). The required heat-transfer coefficient \((\alpha)\) is determined.\(^1\)
Tables No. 1 and No. 2 give the calculated values of \(\varphi\) as a function of the magnitude of the Fourier and Biot criteria, for heating–cooling of plane and cylindrical bodies; here the heating is two-sided (for a plane body) and, for a cylindrical body, heating over the entire surface (see Tables No. 1 and No. 2).
From equation (17), graphical constructions have been made in coordinate axes: \(\varphi\)—ordinate axis, \(Fo\)—abscissa axis.
Analyzing the tabular data and the graphical constructions on the \(Fo—Bi\) diagram, we find that the degree of uniformity of heating–cooling depends predominantly on the value of the Fourier criterion and depends very little on the value of the Biot criterion.
In calculating the tabular data and in constructing the diagram, the values of the roots \(\delta_k\) needed for calculating \(\Phi_o(Bi,Fo)\) and \(\Phi_m(Bi,Fo)\), solved by H. Gröber, were used (see Table 2 on p. 37 in the book by H. Gröber and Erk, Fundamentals of the Theory of Heat Transfer).
In the lower half of the diagram, curves of the temperature potential are plotted as functions of the Fourier and Biot criteria according to the equation
\[ \psi=\frac{t_m}{t_b}=\varphi\left[1-\Phi_o(Bi,Fo)\right]. \tag{18} \]
The term “temperature potential” is defined as the ratio of the temperature on the axis of the body being heated or cooled to the temperature of the heating–cooling medium. To derive equation (18), let us solve equations (15) and (17) jointly. From equation (15)
\[ t_m=t_b-t_b\Phi_m(Bi,Fo)=\left[1-\Phi_m(Bi,Fo)\right]t_b \]
we substitute into (17); then
\[ \varphi=\frac{t_m}{t_b\left[1-\Phi_o(Bi,Fo)\right]}, \]
denoting \(\psi=\dfrac{t_m}{t_b}\), we obtain
\[ \psi=\frac{t_m}{t_b}=\varphi\left[1-\Phi_o(Bi,Fo)\right]. \tag{18} \]
Equation (18) makes it possible to determine the temperature potential, and consequently also the temperature of the heating or cooling medium, from the required degree of uniformity of heating–cooling and from the values of the criteria.
The typical problems solved with the aid of equation (18) are as follows:
A. Given: the degree of uniformity of heating \((\varphi)\), the heat-transfer coefficient \((\alpha)\), and the heating–cooling time \((\tau)\). The required temperature of the heating–cooling medium \((t_b)\) is determined.
B. Given: the temperature of the heating–cooling medium \((t_b)\), the heat-transfer coefficient \((\alpha)\), and the heating–cooling time \((\tau)\). The degree of uniformity of heating–cooling \((\varphi)\) is determined.
C. Given: the temperature of the heating–cooling medium \((t_b)\), the degree of uniformity of heating–cooling \((\varphi)\), and the heat-transfer coefficient \((\alpha)\). The duration of heating–cooling \((\tau)\) is determined.
Tables No. 3 and No. 4 give the calculated values of the temperature potential \(\psi\) as a function of the magnitude of the Fourier and Biot criteria
\(^1\) Note. Both here and below, for brevity, we shall understand the heat-transfer coefficient in an extended sense, i.e., either as the conductance coefficient, or as the heat-transmission coefficient, or as the heat-transfer coefficients proper, depending on which boundary temperatures are taken into account.
when heating-cooling flat and cylindrical bodies under the same heating-cooling conditions as were adopted in calculating the tabular data (Tables Nos. 1, 2, 3, and 4).
Let us give an example of calculating the functions of the Biot and Fourier criteria and the coordinates of the points of the diagram curves.
From equations (12 and 14)
\[ \Phi_{o}(Bi,Fo)=\sum_{k=1}^{k=\infty} 2 \, \frac{\sin \delta_k}{\delta_k+\sin \delta_k \cos \delta_k}\, e^{-\delta_k^2 \frac{a\tau}{x^2}}\cos \delta_k . \]
Let the duration of heating be \(\tau=16,8\) hours;
the thermal diffusivity of the body being heated \(a=0,00475\ \text{m}^2/\text{hour}\);
half the thickness of the body being heated \(x=0,2\ \text{m}\);
then the criterion
\[ Fo=\frac{a\tau}{x^2}, \]
\[ Fo=\frac{0,00475\cdot 16,8}{(0,2)^2}=2,0. \]
The wall thickness \(S_{cm}=0,12\ \text{m}\);
the coefficient of thermal conductivity of the wall material
\[ \lambda=1,45\ \text{kcal}/\text{m hour}^{\circ}\text{C}; \]
the conductance of the wall material
\[ \beta=\frac{\lambda_{cm}}{\delta_{cm}}=\frac{1,45}{0,12}=12,0\ \text{kcal}/\text{m}^2\text{hour}^{\circ}\text{C}; \]
the coefficient of thermal conductivity of the body being heated
\[ \lambda=1,6\ \text{kcal}/\text{m hour}^{\circ}\text{C}; \]
therefore the Biot criterion
\[ Bi=\frac{\beta}{\lambda}\,x, \]
\[ Bi=\frac{12,0}{1,6}\cdot 0,2=1,5. \]
From Table 2, p. 37, in the book by H. Gröber and Erk, Fundamentals of the Theory of Heat Transfer, we find, by interpolation, the values of the roots \(\delta_k\) as a function of the Biot criterion.
For \(Bi=1,5\): \(\delta_{k_1}=0,925\); \(\delta_{k_2}=3,525\); \(\delta_{k_3}=6,5\); \(\delta_{k_4}=9,573\) and \(\delta_{k_5}=12,69\).
Determine the values of the circular functions (in radians):
\[ \begin{aligned} &\text{for } \delta_{k_1}=0,925 \rightarrow \sin \delta_{k_1}=0,79860;\ \cos \delta_{k_1}=0,60182\\ &\text{for } \delta_{k_2}=3,525 \rightarrow \sin \delta_{k_2}=0,37080;\ \cos \delta_{k_2}=0,91640\\ &\text{for } \delta_{k_3}=6,50 \rightarrow \sin \delta_{k_3}=0,21823;\ \cos \delta_{k_3}=0,97590\\ &\text{for } \delta_{k_4}=9,573 \rightarrow \sin \delta_{k_4}=0,14944;\ \cos \delta_{k_4}=0,98877\\ &\text{for } \delta_{k_5}=12,69 \rightarrow \sin \delta_{k_5}=0,11971;\ \cos \delta_{k_5}=0,99281 \end{aligned} \]
Let us calculate the complex
\[ e^{-\delta_k^2 \frac{a\tau}{x^2}}=e^{-\delta_k^2 Fo}=e^{-y}, \]
where \(y=-\delta_k^2 Fo\).
\[ \begin{aligned} y_1&=-0.925^2\cdot 2=-1.71\\ y_2&=-3.525^2\cdot 2=-25\\ y_3&=-6.5^2\cdot 2=-84\\ y_4&=-9.573^2\cdot 2=-182\\ y_5&=-12.69^2\cdot 2=-320 \end{aligned} \]
Let us compute \(e^{-y_1}=e^{-1.71}=0.18095\) (see Hütte, p. 39, table 4, vol. 1, 15th ed.).
It is of no practical use to compute the remaining values \(e^{-y_2}\), \(e^{y_3}\), etc.; therefore, for values of the Fourier criterion greater than 2, in determining \(\Phi(B_i,F_o)\) one may restrict oneself to only the first term of the series; for values of the Fourier criterion in the range from 0.05 to 1, in determining \(\Phi(B_i,F_o)\) one may restrict oneself to three terms of the series.
Substituting all these values into the expression for the function \(\Phi(Bi,Fo)\), we have
\[ \Phi^o(Bi,F_o=2)= \frac{\sin \delta_k}{\delta_{k_1}+\sin\delta_{k_1}\cos\delta_{k_1}}\, e^{-\delta_{k_1}^{\,2}\frac{a\tau}{x^2}}\, \cos\delta_{k_1} \]
\[ \Phi_o(Bi,Fo)=2\, \frac{0.79800}{0.925+0.79860\cdot 0.60182} \cdot 0.18096\cdot 0.60182 \]
\[ \Phi_o(Bi,Fo)=0.102 \]
To compute \(\Phi_m(Bi,Fo)\), we note the expression for this function from equations (13 and 15)
\[ \Phi_m=(BiFo)=\sum_{k=1}^{k=\infty} 2\,\frac{\sin\delta_k}{\delta_k+\sin\delta_k\cos\delta_k}\, e^{-\delta_k^{\,2}\frac{a\tau}{x^2}}. \]
Comparing \(\Phi_m(Bi,F_o)\) and \(\Phi_o(Bi,F_o)\), we observe that \(\Phi_m(Bi,F_o)\) differs from \(\Phi_o(B_iF_o)\) by the absence of the factor \(\cos\delta_k\).
Then
\[ \frac{\Phi_m(Bi,Fo)}{\Phi_o(Bi,Fo)}=\frac{1}{\cos\delta_k}, \]
whence
\[ \Phi_m(Bi,Fo)=\frac{\Phi_o(Bi,Fo)}{\cos\delta_k}. \]
In our case
\[ \Phi_m(Bi,F_o)=\frac{0.102}{0.60182}=0.169. \]
In the same way, the values \(\Phi_o(Bi,Fo)\) and \(\Phi_m(Bi,Fo)\) were computed for values of the criteria \(Bi\) from 0.2 to 20.0 and for values of the criteria \(Fo\) from 0.05 to 10.0, both for a plane body and for a cylindrical one. The curves in the book Introduction to the Theory of Heat Transfer, 1929, pp. 52, 53, figs. 20, 21, 22, 23, as well as the tabular data of W. Heidigenstedt in the book Thermal Engineering Calculations, 1937, pp. 109, 110, 111, tables 56, 57, 59, 60, were of great help in the numerous calculations; unfortunately, however, there are many misprints in W. Heidigenstedt’s tabular data.
By substituting the values \(\Phi_o(Bi,Fo)\) and \(\Phi_m(Bi,Fo)\) into the equation for the degree of uniformity of heating, which has the form
\[ \varphi=\frac{1-\Phi_m(Bi,Fo)}{1-\Phi_o(Bi,Fo)} \]
and into the equation for the temperature potential, which has the form
\[ \psi=\varphi[1-\Phi_o(Bi,Fo)], \]
for the particular example under consideration the degree of uniformity of heating of the body will be
\[ \varphi=\frac{1-\Phi_m(Bi,Fo)}{1-\Phi_o(Bi,Fo)} =\frac{1-0.169}{1-0.102}=0.94, \]
i.e., the temperature on the axis of the body being heated will be \(0.94\) of the temperature at the surface of the body
\[ t_m=\varphi t_o;\quad t_m=0.94; \]
the temperature potential will be
\[ \psi=\varphi[1-\Phi_o(Bi;Fo)]=0.94(1-0.102)=0.86, \]
i.e., the temperature at the heating surface (wall) facing the heat carrier (gas), for the given conductivity of the wall, is 1.16 times higher than the temperature on the axis of the body being heated, since
\[ \psi=\frac{t_m}{t_b};\quad t_b=\frac{t_m}{\psi}=\frac{1}{0.86}\,t_m=1.16\,t_m, \]
the temperature potential between the heating surface (wall) facing the heat carrier and the surface of the body being heated will be
\[ \psi=\frac{t_m}{t_b}=\varphi\,\frac{t_o}{t_b}. \]
In our case, therefore, the temperature at the heating surface (wall) facing the heat carrier (gas) is 1.10 times higher than the temperature at the surface of the body being heated, since
\[ t_b=\frac{\varphi}{\psi}\,t_o=\frac{0.940}{0.860}\,t_o=1.10t_o \]
The joint solution of equations (17) and (18) and the joint use of the upper half (for the uniformity of heating \((\varphi)\)) and the lower half of the diagram (for the temperature potential \((\psi)\)) make it possible to find a rapid and theoretically exact solution to questions related to heating in industrial furnaces. In addition, from the character of the change of the curves in the diagram, one can form a definite idea of the optimal conditions of heating (cooling).
For muffle furnaces, in which the material being heated is in direct contact with the furnace walls and heating occurs due to the thermal conductivity of the walls, the term medium temperature \(t_b\) should be understood as the temperature of the heating surface of the wall facing the combustion chamber. This occurs, for example, in coke ovens. In this case the Biot criterion should be calculated not from the heat-transfer coefficient \(\alpha\), kcal/m²·h·°C, but from the conductivity of the heating surface
\[ \beta=\frac{\lambda_{cm}}{\delta_{cm}}\ \text{kcal}/\text{m}^2\text{h}^\circ\text{C}; \]
here \(\lambda_{cm}\) is the coefficient of thermal conductivity of the material of the furnace wall and \(\delta_{cm}\) is the thickness of the furnace wall (see the preceding example). Otherwise the method of using the diagram is identical with the preceding one (see the diagram).
PRACTICAL USE OF THE DIAGRAM
Below, two examples of using the diagram for calculating the heating of bodies are analyzed; the first example is from the practice of heating furnaces for heating ingots, and the second example is from the practice of coke ovens for coking coal. These examples are taken to illustrate the application of the diagram in the case of direct flame heating of metal and in the case of muffle heating in coking processes.
EXAMPLE 1. ANALYSIS OF THE HEATING OF A STEEL INGOT IN A HEATING FLAME FURNACE.
Determine the temperature in the heating furnace and the duration of heating of a steel ingot of rectangular cross section, 600 mm thick, under the condition that the temperature difference at the center of the ingot and on its surface should not exceed 80°C when the surface temperature of the ingot at the end of heating is 1250°C.
Physical constants of the ingot material:
Thermal conductivity \(\lambda = 25\) kcal/m·h·°C.
Heat capacity \(c = 0.17\) kcal/kg·°C.
Specific weight \(\gamma = 7560\) kg/m³.
Thermal diffusivity
\[ a = \frac{\lambda}{c\gamma}\ \text{m}^2/\text{h}. \]
\[ a = \frac{25}{0.17 \cdot 7560} = 0.0195\ \text{m}^2/\text{h}. \]
Solution: 1) The degree of uniformity of heating required by the condition is
\[ \varphi = \frac{t_m}{t_m}; \]
\[ \varphi = \frac{1250 - 80}{1250} = 0.935. \]
2) To compute the Biot criterion, we take the possible heat-transfer coefficient \(\alpha = 450\) kcal/m²·h·°C; then
\[ Bi = \frac{\alpha}{\lambda} x; \]
\[ Bi = \frac{450}{25} \cdot 0.3 = 5.4. \]
3) Determination of the Fourier criterion—by the required value \(\varphi = 0.935\) and the computed value \(Bi = 5.4\), we find from Table 1 or from the curves on the diagram the value of the Fourier criterion
\[ Fo = 1.3, \]
then the duration of heating of the ingot is determined from the equation
\[ F_0 = \frac{a\tau}{x^2}, \]
hence
\[ \tau = Fo \frac{x^2}{a}; \]
\[ \tau = 1.3 \frac{0.3^2}{0.0195} = 6 \text{ hours}. \]
4) Determination of the furnace temperature. The previously found value of the Biot criterion \((Bi = 5.4)\) and of the Fourier criterion \((Fo = 1.3)\) corresponds, from Table 3 or from the curves on the diagram, to the temperature potential at the end of heating
\[ \psi = \frac{t_m}{t_b} = 0.87; \]
then the furnace temperature must be
\[ t_b = \frac{t_m}{\psi}, \]
\[ t_b = \frac{1170}{0.87} = 1350^\circ \mathrm{C}. \]
5) Verification of the heat-transfer coefficient. The mean temperature of the ingot is
\[ t_{\mathrm{cp.}} = \frac{t_0^1 + t_0}{2}, \]
\[ t_{\mathrm{cp}} = \frac{20 + 1250}{2} = 635^\circ \mathrm{C}. \]
The heat-transfer coefficient by radiation is
\[ \alpha_s = \frac{ c \left[ \left(\frac{T_B}{100}\right)^4 - \left(\frac{T_{\mathrm{cp}}}{100}\right)^4 \right] }{ t_b - t_{\mathrm{cp}} }, \]
\[ \alpha_s = \frac{ 4.8 \left(\frac{1350 + 273}{100}\right)^4 - \left(\frac{635 + 273}{100}\right)^4 }{ 1350 - 635 } \]
\[ \alpha_s = 425 \ \text{kcal}/\text{m}^2 \text{ hour}^\circ\mathrm{C}. \]
The heat-transfer coefficient by convection is
\[ \alpha_k = 5.3 + 3.6 v, \]
taking \(v = 5 \ \text{m/sec}\), we obtain
\[ \alpha_k = 5.3 + 3.6 \cdot 5 = 23.3 \ \text{kcal}/\text{m}^2 \text{ hour}^\circ\mathrm{C}. \]
Consequently, the total heat-transfer coefficient is
\[ \alpha = \alpha_s + \alpha_k, \]
\[ \alpha = 425 + 23.3 = 448.3 \ \text{kcal}/\text{m}^2 \text{ hour}^\circ\mathrm{C}. \]
For the calculation, however, the adopted total heat-transfer coefficient was
\[ \alpha = 450 \ \text{kcal}/\text{m}^2 \text{ hour}^\circ\mathrm{C}. \]
In view of the insignificant discrepancy between the value of the overall heat-transfer coefficient adopted and the one calculated, there is no need to recalculate. In the case of large discrepancies it would be necessary to recalculate the criteria \(Bi\) and \(Fo\) and the subsequent computations until an admissible difference is obtained.
EXAMPLE 11. ANALYSIS OF THE DURATION OF COKING (HEATING) OF COAL IN A COKE OVEN
A. Determine the degree of uniformity of heating and the temperature on the axis of the coke “cake” in the coking chamber, under the following conditions: 1) thickness of the coal charge \(400\ \text{mm}\); 2) thickness of the silica-brick furnace wall \(120\ \text{mm}\); 3) duration of the coking period 12 h 20 min; 4) temperature of the wall surface of the furnace on the side of the combustion chamber \(1300^\circ\text{C}\).
Physical constants:
Mean coefficient of thermal conductivity of coal-coke
\[
\lambda = 1.55\ \text{kcal}/\text{m}^2\text{ h}^{\circ}\text{C}.
\]
Heat capacity of coal-coke
\[
c = 0.45\ \text{kcal}/\text{kg}^{\circ}\text{C}
\]
Bulk weight of coal-coke
\[
\gamma = 700\ \text{kg}/\text{m}^3.
\]
Thermal diffusivity
\[ a=\frac{\lambda}{c\gamma}=\frac{1.5}{700\cdot 0.45}=0.00475\ \text{m}^2/\text{h}. \]
Mean coefficient of thermal conductivity of the silica-brick walls
\[ \lambda_{cm}=1.151\ \text{kcal}/\text{m}^2\text{ h}^{\circ}\text{C}; \]
conductance of the silica-brick walls
\[ \beta=\frac{\lambda_{cm}}{\delta_{cm}}=\frac{1.51}{0.12}=12.6\ \text{kcal}/\text{m}^2\text{ h}^{\circ}\text{C}. \]
Solution: 1) Determination of the Fourier criterion from the specified duration of coking
\[ Fo=a\frac{\tau}{x^2}, \]
\[ Fo=0.00475\frac{12.3}{(0.2)^2}=1.465. \]
2) Determination of the Biot criterion from the conductance of the wall
\[ Bi=\frac{\beta}{\lambda}x; \]
\[ Bi=\frac{12.6}{1.5}\cdot 0.2=1.68. \]
3) Determination of the temperature potential from the values of the criteria
(see Table No. 3 and the lower half of the diagram for a plane body)
\[ \psi=\frac{t_m}{t_b}=0.792. \]
4) Determination of the temperature on the axis of the coke “cake”
\[ t_m=\psi t_b \]
\[ t_m=0.792-1300=1030^\circ\text{C}. \]
5) Determination of the degree of uniformity of heating and of the temperature at the surface of the coke “cake” (see Table 1 and the upper half of the diagram)
\[ \varphi=\frac{t_m}{t_u}=0.91. \]
Hence
\[ t_0=\frac{t_m}{\varphi}=\frac{1030}{0.91}=1133^\circ\mathrm{C}. \]
B. Determine the influence of the incandescence temperature of the furnace walls on the duration of coking (heating), if the incandescence temperature of the furnace walls varies from \(1200^\circ\mathrm{C}\) to \(1300^\circ\mathrm{C}\) and to \(1400^\circ\mathrm{C}\) \((t_b)\), under the condition that the thickness of the coal charge in the furnace is 400 mm and the temperature on the axis of the coke “cake” by the end of coking must be
\[ t_m=950^\circ\mathrm{C}. \]
We shall compile the specified and calculated quantities in Table 5.
Table No. 5
Duration of coking (heating) as a function of the incandescence temperature of the furnace walls
| No. | Quantity | Dimension | For furnace-wall incandescence temperature 1200°C | For furnace-wall incandescence temperature 1300°C | For furnace-wall incandescence temperature 1400°C | Method of calculation |
|---|---|---|---|---|---|---|
| 1 | Width of furnace charge | \(2x\) mm | 400 | 400 | 400 | Given |
| 2 | Thickness of furnace walls | mm | 120 | 120 | 120 | Given |
| 3 | Coefficient of thermal conductivity of furnace walls | \(\lambda\ \dfrac{\mathrm{kcal}}{m\,\mathrm{h}\,^\circ\mathrm{C}}\) | 1.45 | 1.48 | 1.51 | By the formula \(\lambda=0.7+0.000325\,(t_1+t_2)\) |
| 4 | Conductance of furnace walls | \(\beta\ \dfrac{\mathrm{kcal}}{m^2\,\mathrm{h}\,^\circ\mathrm{C}}\) | 12.1 | 12.35 | 12.6 | By the formula \(\beta=\dfrac{\lambda_{st}}{\delta_{st}}\) |
| 5 | Coefficient of thermal conductivity of coal-coke | \(\lambda\ \dfrac{\mathrm{kcal}}{m\,\mathrm{h}\,^\circ\mathrm{C}}\) | 1.5 | 1.5 | 1.5 | Terres and Baum |
| 6 | Biot criterion | \(Bi\) | 1.615 | 1.65 | 1.68 | \(Bi=\dfrac{\lambda}{\beta}\,x\) |
| 7 | Temperature potential | \(\psi\) | 0.79 | 0.73 | 0.678 | \(\psi=\dfrac{t_m}{t_b}\) |
| 8 | Fourier criterion | \(Fo\) | 1.91 | 1.60 | 1.40 | From Table No. 3 or from the diagram |
| 9 | Degree of uniformity of heating | \(\varphi\) | 0.90 | 0.87 | 0.84 | From Table No. 1 and from the diagram |
| 10 | Temperature at the surface of the coke cake | \(t_0\ ^\circ\mathrm{C}\) | 1052 | 1095 | 1138 | By the formula \(\varphi=\dfrac{t_m}{t_0};\quad t_0=\dfrac{t_m}{\varphi}\) |
| 11 | Thermal diffusivity of coal-coke | \(\dfrac{m^2}{\mathrm{h}}\) | 0.00475 | 0.00475 | 0.00475 | By the formula \(\alpha=\dfrac{\lambda}{c\gamma}\) \(\lambda=1.5\) \(c=0.45\) \(\gamma=700\) |
| 12 | Duration of coking (heating) | \(\tau\) h | 16.1 | 13.5 | 11.8 | From the Fourier criterion: \(Fo=\dfrac{\alpha\tau}{x^2}\) \(\tau=\dfrac{x^2}{\alpha}Fo\) |
Conclusions: With an increase in the heating temperature of the furnace walls on the side of the heating flues, the duration of coking (heating) is considerably reduced; the nonuniformity of the temperature distribution across the thickness of the coal–coke increases.
B. To determine the effect of the conductance of the furnace walls on the duration of coking-heating.
Let us formulate the problem as follows:
“How does the duration of the coking (heating) period vary with a change in the conductance of the furnace walls, provided that the thickness of the charge in the furnace is 400 mm, the temperature of the furnace walls on the side of the heating flues is 1300°C, and the temperature on the axis of the coke cake must be 950°C by the end of coking.”
In the preceding first calculation, in addition to the main problem posed, it was incidentally found that if the wall material is one and the same (silica), then the change in its conductance with a change in the temperature of the wall flue is insignificant: \(\beta' = 12.1\); \(\beta'' = 12.35\) and \(\beta''' = 12.6\); \(t_{cm} = 1200^\circ\mathrm{C}\); \(t_{cm} = 1300^\circ\mathrm{C}\); \(t_{cm} = 1400^\circ\mathrm{C}\).
Therefore, a substantial change in conductance is possible on the condition that either, first, the wall thickness changes considerably, or, second, the coefficient of thermal conductivity of the wall material differs considerably from those used previously. Let us compare the conductances of furnace walls made of fireclay, silica, and magnesite refractory bricks. Let in all three cases the thickness of the wall material be the same—120 mm; but in one case the wall material is fireclay, for which the coefficient of thermal conductivity is \(\lambda_{cm} = 1.2\); in another case the wall material is silica brick, \(\lambda_{cm} = 1.48\); and in the third case magnesite brick, for which the coefficient of thermal conductivity is \(\lambda_{cm} = 3.5\).
Then the conductances are
for the fireclay wall
\[ \beta'=\frac{\lambda_{cm}}{\delta_{cm}}=\frac{1.2}{0.12}=10\ \text{kcal}/\text{m}^2\ \text{h}^{\circ}\mathrm{C}, \]
for the silica wall
\[ \beta''=\frac{1.48}{0.12}=12.35\ \text{kcal}/\text{m}^2\ \text{h}^{\circ}\mathrm{C}, \]
for the magnesite wall
\[ \beta'''=\frac{3.5}{0.12}=29.2\ \text{kcal}/\text{m}^2\ \text{h}^{\circ}\mathrm{C}. \]
We shall place the specified and calculated quantities in a table.
Conclusions: With an increase in the conductance of the wall material, the duration of coking (heating) is reduced considerably, while the nonuniformity of the temperature distribution in a thick layer of coal–coke (for example, 400 mm) increases.
We draw attention to the fact that changing the conductance of the wall material is possible not only by selecting one or another material according to its coefficient of thermal conductivity, but also by changing the thickness of the material.
EFFECT OF HEAT-TRANSFER COEFFICIENTS IN THE HEATING FLUES ON THE DURATION OF COKING
To determine the effect of the heat-transfer coefficients in the heating flues on the duration of coking, it is necessary to calculate the Biot criterion not from the wall conductance alone, but from the incomplete heat-transfer coefficient, i.e.,
\[ Bi=\frac{k}{\gamma}x \]
Table No. 6
Duration of coking (heating) as a function of the conductivity of the furnace wall material.
| No. | Quantity | Dimension | For conductivity of the wall material β′=10 | For conductivity of the wall material β″=12.35 | For conductivity of the wall material β‴=29.2 | Method of calculation |
|---|---|---|---|---|---|---|
| 1 | Width of furnace charge | 2·x mm | 400 | 400 | 400 | Given |
| 2 | Coefficient of thermal conductivity of coal-coke | \(\lambda\ \dfrac{\mathrm{kcal}}{\mathrm{m}\,\mathrm{h}\,^\circ\mathrm{C}}\) | 1.5 | 1.5 | 1.5 | Terres and Baum |
| 3 | Biot criterion | Bi | 1.34 | 3.9 | 3.9 | \(\mathrm{Bi}=\dfrac{\beta}{\lambda}\,x\) |
| 4 | Temperature potential | \(\psi\) | 0.73 | 0.73 | 0.73 | \(\psi=\dfrac{t_m}{t_b}\) |
| 5 | Fourier criterion | Fo | 1.74 | 1.60 | 0.95 | According to Table No. 3 or from the diagram |
| 6 | Degree of uniformity of heating | \(\varphi\) | 0.875 | 0.876 | 0.796 | According to Table No. 1 or from the diagram |
| 7 | Temperature on the surface of the coke cake | \(S_0^\circ\mathrm{C}\) | 1085 | 1095 | 1193 | According to the formula \(t_\mathrm{e}=\dfrac{t_m}{\varphi}\) |
| 8 | Duration of (coking) heating | \(\tau\) h | 14.6 | 13.5 | 8.0 | From the Fourier criterion \(\tau=\dfrac{x^2}{a}\,\mathrm{Fo}\) |
where
\[ k=\frac{1}{\dfrac{1}{\alpha^k+\alpha^s}+\dfrac{\delta_{cm}}{\lambda_{cm}}} \tag{19} \]
or, substituting,
\[ \frac{S_{cm}}{\lambda}=\frac{1}{\beta};\quad K=\frac{1}{\dfrac{1}{\alpha^k+\alpha^s}+\dfrac{1}{\beta}}. \]
The heat-transfer coefficient \(\alpha=\alpha^k+\alpha^s\) may vary within rather wide limits. The average heat-transfer coefficient in the heating system of coke ovens may be estimated as \(\alpha=40\ \mathrm{kcal}/\mathrm{m}^2\,\mathrm{h}\,^\circ\mathrm{C}\), but in the zone of direct combustion of the heating gas (in the zone of direct heat release) the value of the heat-transfer coefficient increases to
\[ \alpha \sim = 120\ \mathrm{kcal}/\mathrm{m}^2\,\mathrm{h}\,^\circ\mathrm{C}. \]
Let us formulate the problem as follows: “How does the duration of the coking (heating) period change with variation of the heat-transfer coefficients, under the condition that the thickness of the charge in the furnace is 400 mm, the temperature of the heating gaseous medium is 1450°C, and the temperature on the axis of the coke cake must be 950°C by the end of coking, while the conductivity of the walls in one case is \(\beta'=12.35\) and in the other case \(\beta''=29.2\).”
The given and calculated quantities are brought together in a table (Table 7).
Conclusions: 1) An increase in the heat-transfer coefficients at unchanged conductivity of the furnace walls and temperature of the heating gases has an extremely small effect on the duration of heating. An increase in the heat-transfer coefficient...
Table No. 7
Duration of coking (heating) as a function of the heat-transfer coefficients of the gas pipe.
| No. | Quantity | Dimension | For wall conductivity and heat-transfer coefficient, $\beta'=12.35$, $\alpha=40$ | For wall conductivity and heat-transfer coefficient, $\beta'=12.35$, $\alpha=120$ | For wall conductivity and heat-transfer coefficient, $\beta''=29.2$, $\alpha=40$ | For wall conductivity and heat-transfer coefficient, $\beta''=29.2$, $\alpha=120$ | Method of calculation |
|---|---|---|---|---|---|---|---|
| 1 | Heat-transfer coefficient | $K=\dfrac{kcal}{m^2\,hour\,^\circ C}$ | 9.44 | 11.2 | 16.9 | 23.5 | $K=\dfrac{1}{\dfrac{1}{\alpha}+\dfrac{1}{\beta}}$ |
| 2 | Biot criterion | $Bi$ | 1.26 | 1.49 | 2.25 | 3.14 | $Bi=\dfrac{K}{\lambda}x$ |
| 3 | Temperature of the heating gases | $t_2\,^\circ C$ | 1450 | 1450 | 1550 | 1550 | Given |
| 4 | Temperature potential | $\psi$ | 0.655 | 0.655 | 0.613 | 0.613 | $\psi=\dfrac{t_m}{t_b}$ |
| 5 | Fourier criterion | $Fo$ | 1.465 | 1.46 | 1.035 | 0.91 | From Table No. 3 or from the diagram |
| 6 | Degree of uniformity of heating | $\varphi$ | 0.837 | 0.826 | 0.766 | 0.731 | From Table No. 3 or from the diagram |
| 7 | Temperature of the surface of the coke cake | $t_0\,^\circ C$ | 1136 | 1150 | 1240 | 1302 | $t_0=\dfrac{t_m}{\varphi}$ |
| 8 | Duration of heating (coking) | $\tau\,hour$ | 12.35 | 12.30 | 8.80 | 7.65 | From the Fourier criterion $\tau=\dfrac{x^2}{a}Fo$ |
| 9 | Temperature of the heating surface facing the heat carrier | $t\,^\circ C$ | 1380 | 1410 | 1420 | 1500 | From the temperature potential between the body and the wall |
…of heat transfer threefold, $\alpha=40$ and $\alpha=120$) reduces the duration of heating by hundredths of an hour.
2) Increasing the conductivity of the furnace walls by a factor of two (approximately—since $\beta'=12.35$ and $\beta''=29.2$) considerably reduces the duration of heating (coking): with $\beta'=12.35$ hours, at $\tau=12.35$, and $\tau=8.8$ hours with $\beta''=29.2$; however, the influence of increasing the heat-transfer coefficients is also insignificant at $\alpha=40$—$\tau=8.8$ hours—and at $\alpha=120$, $\tau=7.65$.
Consequently, the most substantial influence is exerted by the conductivity of the furnace walls, which is also a characteristic feature of the heating system of a coke oven as a muffle-type furnace.
DURATION OF COKING (HEATING) AS A FUNCTION OF THE THICKNESS OF THE COAL-COKE IN THE OVEN
Changing the thickness of the coal charge in the oven (changing the width of the coking chamber) entails a change in the internal conditions of coking (heating), and entails a change in the Fourier criterion. Analysis of the duration of coking as a function of the charge thickness makes it possible to reveal the importance of the problem of coking (heating) in a thin layer.
Let us formulate the problem as follows: “How does the duration of the coking (heating) period change with a decrease in the thickness of the coal-coke being charged—”
can be in the furnace. Condition: the temperature by the end of the coking period on the axis must be 950°C; the furnace wall, of dinas brick, has conductivity \(\beta = 12.35\ \mathrm{cal}/\mathrm{m}^2\mathrm{h}^{\circ}\mathrm{C}\); the temperature of the surface of the furnace walls is 1300°C (on the side of the verticals), and the thickness of the coal–coke charge is 400 mm, 300 mm, 200 mm, and 100 mm.
We shall put the specified and calculated values into a table (Table No. 8).
Table No. 8
Duration of heating (coking) as a function of the thickness of the coal charge in the furnace
| No. | Quantity | Dimension | For charge thickness 400 mm | For charge thickness 300 mm | For charge thickness 200 mm | For charge thickness 100 mm | Method of calculation |
|---|---|---|---|---|---|---|---|
| 1 | Biot criterion . . . . | \(Bi\) | 1.65 | 1.23 | 0.825 | 0.412 | \(Bi=\dfrac{\beta}{\lambda}x\) |
| 2 | Temperature potential . . . . | \(\psi\) | 0.73 | 0.73 | 0.73 | 0.73 | \(\psi=\dfrac{t_m}{t_b}\) |
| 3 | Fourier criterion . . . . | \(Fo\) | 1.60 | 1.75 | 2.29 | 3.77 | From Table No. 3 or from the diagram |
| 4 | Degree of uniformity of heating . . . . | \(\varphi\) | 0.876 | 0.880 | 0.90 | 0.94 | From Table No. 1 or from the diagram |
| 5 | Temperature on the surface of the coke “pie” . . . . | \(t_0^\circ\mathrm{C}\) | 1085 | 1082 | 1054 | 1010 | \(t_0=\dfrac{t_m}{\varphi}\) |
| 6 | Duration of coking (heating) . . . . | \(\tau_{\text{h}}\) | 13.5 | 8.3 | 4.82 | 1.99 | From the Fourier criterion \(\tau=\dfrac{x^2}{a}Fo\) |
Conclusions: 1) With a decrease in the thickness of the charge, all other conditions being equal, the Biot criteria decrease, while the Fourier criteria increase; this entails a considerable reduction in the heating time. The decrease in heating (coking) time is proportional to the square of the ratio of thicknesses, the proportionality coefficients being different for different limiting ratios. Thus, for a thickness of 400–300 mm,
\[ \frac{\tau'}{\tau''}=0.915\left(\frac{x_1}{x_2}\right)^2; \]
for a thickness of 300–200 mm,
\[ \frac{\tau'}{\tau''}=0.715\left(\frac{x_1}{x_2}\right)^2; \]
for a thickness of 200–100 mm,
\[ \frac{\tau'}{\tau''}=0.608\left(\frac{x_1}{x^2}\right)^2. \]
2) A decrease in the thickness of the charge entails, in addition to a decrease in the heating time, an equalization of the temperature on the axis and on the surface. Compare: for 400 mm the temperature difference is 135°C, and for 100 mm the temperature difference is 60°C.
The detailed analysis performed of the duration of heating of coal (coking) lays claim to approximate quantitative accuracy, since in the analysis the absence of chemical transformations during coking and the constancy of the physical constants (averaged values) were taken as self-evident.
ON THE CONSTRUCTION OF APPROXIMATE FORMULAS FOR CALCULATING THE DURATION OF HEATING–COOLING PROCESSES OF SOLIDS
The physico-mathematical theory of the nonstationary temperature field, expressed by the Fourier and Gröber equations, as well as the criterion of heating–cooling of solids—the Fourier criterion—show that the duration of heating–cooling processes is a quadratic dependence on the thickness of the solid. Therefore, obviously, this quadratic dependence must be reflected in the construction of approximate formulas. In addition, in constructing approximate formulas for calculating the heating–cooling of solids, the degree of uniformity of heating admissible for the given body must be reflected. In the absence of such a prior stipulation, approximate formulas have no meaning.
Analyzing the influence of the Fourier and Biot criteria on the degree of uniformity of heating (Tables 1 and 3 and the diagram), we note.
1) For one and the same value of the Fourier criterion, the influence of the Biot criterion on the degree of uniformity is insignificant.
For example, at \(Fo = 2\) and \(Bi = 0.20\), \(\varphi = 0.88\); \(Fo = 2\), \(Bi = 10\), \(\varphi = 0.990\), i.e., when the Biot criterion is increased 50-fold, \(\varphi\) increases by only 13%. Moreover, with an increase in the Biot criterion, \(\varphi\) also increases.
2) For one and the same value of the Biot criterion, the influence of the Fourier criterion, within the range of small values of the latter, on the uniformity of heating is considerable. Compare, for example, the value of \(\varphi\) for \(Bi = 2\) and for \(Fo = 1\) (\(\varphi = 0.845\)) and for \(Fo = 0.4\) (\(\varphi = 0.61\)).
3) For values of the Biot criterion close to 1.5, practically permissible temperature equalization occurs at values of the Fourier criterion approximately equal to 2; for values of the Biot criterion close to 0.6, practically permissible temperature equalization occurs at values of the Fourier criterion approximately equal to 2.5.
This makes it possible to compose equations of the form
\[ \tau = f(x^2), \]
\[ \text{at } \begin{cases} Bi = \mathrm{const},\\ \varphi = \mathrm{const}, \end{cases} \]
i.e., to determine the duration of heating–cooling as a function of one variable—the thickness of the charge. Applying such a method to a specific case—the duration of coking (heating) of coal in a coke oven—we obtain that the equation of the heating (coking) time for a charge thickness of 400–300 mm has the form
\[ Fo = 2 = \frac{a}{x^2}\tau \]
or
\[ \tau = 2\frac{x^2}{a}. \tag{20} \]
For a charge thickness of 200–100 mm
\[ Fo = 2.5 = \frac{a}{x^2}\tau \]
or
\[ \tau = 2.5\frac{x^2}{a}; \tag{21} \]
Table No. 1. Degrees of equal density of the grid for two-sided surrounding-cooling of an unbounded slab
$$ \varphi = \frac{1-\Phi_m(\mathrm{Bi},Fo)} {1-\Phi_o(\mathrm{Bi},Fo)} $$
| $Fo$ | $\mathrm{Bi}=0.2$ | $\mathrm{Bi}=0.4$ | $\mathrm{Bi}=0.6$ | $\mathrm{Bi}=0.8$ | $\mathrm{Bi}=1$ | $\mathrm{Bi}=1.5$ | $\mathrm{Bi}=2$ | $\mathrm{Bi}=3$ | $\mathrm{Bi}=5$ | $\mathrm{Bi}=10$ | $\mathrm{Bi}=20$ |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0.05 | 0.183 | 0.186 | 0.200 | 0.201 | 0.202 | 0.203 | 0.208 | 0.208 | 0.215 | 0.228 | 0.240 |
| 0.10 | 0.282 | 0.282 | 0.283 | 0.284 | 0.288 | 0.292 | 0.300 | 0.300 | 0.304 | 0.330 | 0.343 |
| 0.15 | 0.350 | 0.351 | 0.354 | 0.354 | 0.357 | 0.364 | 0.371 | 0.380 | 0.384 | 0.410 | 0.422 |
| 0.20 | 0.398 | 0.402 | 0.405 | 0.414 | 0.415 | 0.420 | 0.430 | 0.442 | 0.445 | 0.475 | 0.488 |
| 0.40 | 0.552 | 0.565 | 0.570 | 0.570 | 0.572 | 0.598 | 0.610 | 0.624 | 0.635 | 0.682 | 0.747 |
| 0.60 | 0.65 | 0.668 | 0.673 | 0.683 | 0.690 | 0.710 | 0.730 | 0.740 | 0.752 | 0.80 | 0.89 |
| 0.80 | 0.72 | 0.734 | 0.745 | 0.753 | 0.762 | 0.780 | 0.794 | 0.815 | 0.830 | 0.875 | 0.958 |
| 1.00 | 0.78 | 0.80 | 0.81 | 0.812 | 0.813 | 0.833 | 0.845 | 0.866 | 0.880 | 0.900 | 0.920 |
| 1.5 | 0.837 | 0.855 | 0.868 | 0.88 | 0.890 | 0.910 | 0.92 | 0.947 | 0.980 | 0.990 | 0.995 |
| 2.0 | 0.88 | 0.894 | 0.910 | 0.922 | 0.930 | 0.948 | 0.960 | 0.970 | 0.980 | 0.990 | 0.995 |
| 2.5 | 0.905 | 0.920 | 0.938 | 0.945 | 0.955 | 0.970 | 0.973 | 0.986 | 0.99 | 0.999 | 0.999 |
| 3.0 | 0.922 | 0.940 | 0.952 | 0.962 | 0.967 | 0.980 | 0.99 | 0.995 | 0.999 | 1.0 | 1.0 |
| 5.0 | 0.96 | 0.976 | 0.985 | 0.99 | 0.995 | 0.997 | 0.999 | 0.999 | 1.0 | 1.0 | 1.0 |
| 10.0 | 0.988 | 0.995 | 0.997 | 0.998 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 |
Table No. 2
Degree of uniformity of heating-cooling
\[ \varphi=\frac{1-\Phi_m(\mathrm{Bi},Fo)}{1-\Phi_o(\mathrm{Bi},Fo)} \]
For a cylindrical body
| $Fo$ | $\mathrm{Bi}=0,2$ | $0,4$ | $0,6$ | $0,8$ | $1,0$ | $1,5$ | $2,0$ | $3,0$ | $5,0$ | $10$ | $20$ |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0,05 | 0,264 | 0,324 | 0,324 | 0,324 | 0,324 | 0,326 | 0,352 | 0,371 | 0,382 | 0,400 | 0,430 |
| 0,10 | 0,455 | 0,468 | 0,473 | 0,475 | 0,476 | 0,477 | 0,482 | 0,515 | 0,532 | 0,554 | 0,582 |
| 0,15 | 0,558 | 0,566 | 0,568 | 0,576 | 0,577 | 0,578 | 0,582 | 0,634 | 0,636 | 0,660 | 0,688 |
| 0,20 | 0,614 | 0,630 | 0,636 | 0,637 | 0,638 | 0,650 | 0,657 | 0,690 | 0,715 | 0,740 | 0,763 |
| 0,40 | 0,776 | 0,780 | 0,785 | 0,798 | 0,805 | 0,82 | 0,832 | 0,87 | 0,880 | 0,900 | 0,916 |
| 0,60 | 0,842 | 0,856 | 0,860 | 0,873 | 0,88 | 0,897 | 0,907 | 0,931 | 0,946 | 0,965 | 0,973 |
| 0,80 | 0,882 | 0,90 | 0,906 | 0,910 | 0,924 | 0,937 | 0,950 | 0,966 | 0,977 | 0,987 | 0,990 |
| 1,00 | 0,906 | 0,922 | 0,930 | 0,940 | 0,946 | 0,960 | 0,972 | 0,982 | 0,990 | 0,996 | 1,00 |
| 1,50 | 0,941 | 0,956 | 0,965 | 0,973 | 0,980 | 0,990 | 0,992 | 0,999 | 0,999 | 0,999 | 1,00 |
| 2,00 | 0,960 | 0,974 | 0,98 | 0,985 | 0,990 | 0,994 | 0,999 | 1,0 | 1,00 | 1,00 | 1,00 |
| 2,50 | 0,970 | 0,982 | 0,99 | 0,994 | 0,997 | 0,999 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 |
| 3,00 | 0,980 | 0,990 | 0,995 | 0,995 | 0,999 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 |
| 5,00 | 0,991 | 0,998 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 |
| 10,00 | 0,999 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 |
Table No. 3.
For a flat plate with two-sided heating-cooling
Temperature potential
$$ \psi=\frac{t_m}{t_b}=\varphi[1-\Phi_o(\mathrm{Bi}, Fo)] $$
| $Fo$ | $\mathrm{Bi}=0,2$ | 0,4 | 0,6 | 0,8 | 1,0 | 1,5 | 2,0 | 3,0 | 5,0 | 10 | 20 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0,05 | 0,00896 | 0,0211 | 0,0259 | 0,0358 | 0,042 | 0,06 | 0,0763 | 0,099 | 0,115 | 0,177 | 0,214 |
| 0,10 | 0,0190 | 0,0368 | 0,052 | 0,064 | 0,080 | 0,108 | 0,134 | 0,171 | 0,20 | 0,224 | 0,214 |
| 0,15 | 0,032 | 0,0542 | 0,076 | 0,096 | 0,115 | 0,155 | 0,186 | 0,232 | 0,27 | 0,352 | 0,392 |
| 0,20 | 0,037 | 0,071 | 0,0976 | 0,125 | 0,148 | 0,196 | 0,234 | 0,287 | 0,328 | 0,416 | 0,460 |
| 0,40 | 0,072 | 0,134 | 0,185 | 0,225 | 0,260 | 0,340 | 0,396 | 0,465 | 0,520 | 0,687 | 0,691 |
| 0,60 | 0,105 | 0,193 | 0,261 | 0,320 | 0,368 | 0,460 | 0,510 | 0,600 | 0,654 | 0,832 | 0,85 |
| 0,80 | 0,139 | 0,248 | 0,332 | 0,400 | 0,455 | 0,550 | 0,616 | 0,700 | 0,750 | 0,924 | 0,962 |
| 1,00 | 0,178 | 0,298 | 0,396 | 0,470 | 0,530 | 0,635 | 0,700 | 0,774 | 0,820 | 0,937 | 0,945 |
| 1,50 | 0,245 | 0,412 | 0,529 | 0,612 | 0,675 | 0,775 | 0,830 | 0,895 | 0,920 | 0,953 | 0,980 |
| 2,00 | 0,314 | 0,506 | 0,633 | 0,717 | 0,778 | 0,862 | 0,906 | 0,944 | 0,970 | 0,985 | 0,995 |
| 2,50 | 0,376 | 0,585 | 0,715 | 0,792 | 0,845 | 0,915 | 0,936 | 0,970 | 0,993 | 0,999 | 1,000 |
| 3,00 | 0,427 | 0,660 | 0,775 | 0,850 | 0,892 | 0,950 | 0,970 | 0,990 | 1,00 | 1,000 | 1,000 |
| 5,00 | 0,607 | 0,830 | 0,920 | 0,963 | 0,975 | 0,99 | 0,999 | 1,00 | 1,00 | 1,00 | 1,00 |
| 10,00 | 0,846 | 0,970 | 0,980 | 0,99 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 |
Table No. 4
Dimensionless temperature potential
\[ \psi=\frac{t_m}{t_b}=\varphi[1-\Phi_o(\mathrm{Bi}, Fo)] \]
For a cylindrical body.
| \(Fo\) | \(\mathrm{Bi}=0.2\) | 0.4 | 0.6 | 0.8 | 1.0 | 1.5 | 2 | 3 | 5 | 10 | 20 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0.05 | 0.0176 | 0.0342 | 0.048 | 0.0607 | 0.0575 | 0.102 | 0.128 | 0.185 | 0.245 | 0.319 | 0.386 |
| 0.10 | 0.037 | 0.068 | 0.095 | 0.124 | 0.147 | 0.196 | 0.239 | 0.316 | 0.396 | 0.480 | 0.545 |
| 0.15 | 0.0565 | 0.102 | 0.143 | 0.190 | 0.214 | 0.277 | 0.331 | 0.440 | 0.508 | 0.596 | 0.657 |
| 0.20 | 0.0724 | 0.134 | 0.187 | 0.232 | 0.273 | 0.348 | 0.412 | 0.510 | 0.600 | 0.684 | 0.736 |
| 0.40 | 0.141 | 0.250 | 0.338 | 0.411 | 0.472 | 0.570 | 0.646 | 0.748 | 0.82 | 0.875 | 0.905 |
| 0.60 | 0.202 | 0.350 | 0.460 | 0.518 | 0.615 | 0.716 | 0.783 | 0.862 | 0.915 | 0.955 | 0.970 |
| 0.80 | 0.259 | 0.438 | 0.565 | 0.642 | 0.718 | 0.812 | 0.871 | 0.980 | 0.962 | 0.920 | 0.990 |
| 1.00 | 0.318 | 0.514 | 0.646 | 0.732 | 0.794 | 0.880 | 0.924 | 0.960 | 0.982 | 0.985 | 1.00 |
| 1.5 | 0.524 | 0.660 | 0.790 | 0.860 | 0.910 | 0.960 | 0.980 | 0.996 | 0.999 | 1.00 | 1.00 |
| 2.0 | 0.534 | 0.760 | 0.874 | 0.925 | 0.960 | 0.980 | 0.996 | 0.999 | 1.00 | 1.00 | 1.00 |
| 2.5 | 0.612 | 0.835 | 0.925 | 0.960 | 0.981 | 0.995 | 0.998 | 1.00 | 1.00 | 1.00 | 1.00 |
| 3.0 | 0.682 | 0.884 | 0.956 | 0.978 | 0.992 | 0.999 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
| 5.0 | 0.850 | 0.974 | 0.995 | 0.998 | 0.999 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
| 10.0 | 0.979 | 0.999 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
Or, combining equations (20) and (21) into one for large and small charge thicknesses, we obtain
\[ \tau = (2 \div 2.5)\frac{x^2}{a}. \tag{22} \]
If the thermal diffusivity of coal-coke is taken as constant and equal to
\[ a = 0.00475, \]
then the heating-time equation will be
\[ \tau = (421 \div 526)x^2. \tag{23} \]
The calculation according to equation (23) is given in Table No. 9.
Table No. 9
Variation in coking (heating) time as a function of the furnace charge thickness at values of \(\varphi\) approximately equal to 0.9.
| Half of the furnace charge thickness, mm | Heating duration according to the equations, \(\tau = 421x^2\) h | Heating duration according to the equations, \(\tau = 526x^2\) h |
|---|---|---|
| 250 | 26.4 | — |
| 225 | 21.3 | — |
| 200 | 16.8 | — |
| 175 | 12.6 | — |
| 150 | 9.4 | — |
| 125 | 6.6 | — |
| 100 | 4.2 | 5.26 |
| 75 | 2.4 | 2.96 |
| 50 | 1.1 | 1.32 |
If in practice one can be satisfied with the value \(\varphi = 0.86\text{—}0.87\) for a small charge thickness, there is no need to use the equation \(\tau = 526x^2\) for small thicknesses; then only one parabolic equation remains for any thickness,
\[ \tau = 421x^2. \]
CONCLUSION
- The proposed \(F_0—Bi\)-diagram has been constructed on the principle of a combined use of the physico-mathematical theory of heating and the theory of similarity; as a result it is universal.
- Use of the diagram makes it possible to solve essential problems of furnace heat engineering rather quickly, without detriment to accuracy.
- With the aid of the diagram, by means of trial solutions, optimal conditions for heating (cooling) bodies can be selected in the design of furnace installations.
REFERENCES
- H. Gröber and Erk. Fundamentals of the Theory of Heat Exchange. ONTI NKTP, 1936.
- A. A. Gukhman. Physical Foundations of Heat Transfer. ONTI NKTP, 1936.
- M. Adams. Heat Transfer. ONTI NKTP, 1936.