HEATING OF BODIES IN A RADIATING MEDIUM WITH VARIABLE TEMPERATURE

V. V. Salomatov, G. P. Boikov

(Presented by Prof. Dr. G. I. Fuks)

In the practice of heating bodies, one often encounters processes occurring at a variable temperature of the radiating heat source. Observance of the correct heating technology under the indicated heat-exchange conditions requires knowledge of the temperature distribution in the body, which is found as the solution of the following system of differential equations:

\[ \frac{\partial T(\zeta,\tau)}{\partial \tau} = a\left[ \frac{\partial^2 T(\zeta,\tau)}{\partial \zeta^2} + \frac{\xi-1}{\zeta} \frac{\partial T(\zeta,\tau)}{\partial \zeta} \right], \tag{1} \]

\[ T(\zeta,0)=T_0, \tag{2} \]

\[ \frac{\partial T(0,\tau)}{\partial \zeta}=0, \tag{3} \]

\[ \lambda \frac{\partial T(R,\tau)}{\partial \zeta} = \varepsilon_n C_0 \left[ \left(\frac{T_c(\tau)}{100}\right)^4 - \left(\frac{T(R,\tau)}{100}\right)^4 \right]. \tag{4} \]

The solution of such a general problem by ordinary analytical methods is completely impossible because of the nonlinear condition (4), while the method of numerical integration requires a very long time, so that, when a large number of necessary calculations is involved, its practical value is reduced to zero.

In connection with this, it becomes necessary to find approximate methods for calculating the temperature field which, in those cases where an exact solution is impossible, would quickly lead to the goal, even at the cost of a reduction in the accuracy of the calculation.

Transform the boundary condition (4) to the form

\[ T(R,\tau)=T_c(\tau)(1-Z)^{1/4}, \tag{5} \]

where

\[ Z=\frac{10^8 \lambda\, \partial T(R,\tau)/\partial \zeta}{\varepsilon_n C_0 T_c^4(\tau)}. \]

Expanding in a binomial series,

\[ (1-Z)^{1/4}=1-\frac{1}{4}Z-\frac{1\cdot 3}{4\cdot 8}Z^2-\ldots \tag{6} \]

and, in our case restricting ourselves to two terms of the expansion, we obtain

\[ T(R,\tau)=T_c(\tau)-\psi\frac{\partial T(R,\tau)/\partial \xi}{T_c^3(\tau)}. \tag{7} \]

Here

\[ \psi=\frac{10^8}{4}\frac{\lambda}{\varepsilon_n C_0}. \]

As an example we shall give solutions for an infinite plate, an infinite cylinder, and a sphere, when the temperature of the medium changes according to an exponential law

\[ T_c(\tau)=T_k+(T_{c0}-T_k)\exp(-\kappa\tau). \]

The solution is carried out with the aid of the one-sided Laplace integral transform [1]

\[ F(s)=\int_0^\infty f(\tau)e^{-s\tau}\,d\tau \]

and the displacement theorem

\[ F(s-\kappa)=\int_0^\infty f(\tau)e^{-(s-\kappa)\tau}\,d\tau. \]

I. Temperature field of an infinite plate

The problem reduces to solving differential equation (1) with conditions (2), (3), (7) for \(\xi=x\), \(\xi=1\).

Passing to transforms and taking (2), (3) into account, we obtain the solution of equation (1)

\[ \overline{T}(x,s)-\frac{T_0}{s}=A\,\operatorname{ch}\sqrt{\frac{s}{a}}\,x. \tag{I.1} \]

The constant \(A\) is found from the transformed boundary condition (7)

\[ \begin{aligned} A\Bigg[& \psi\sqrt{\frac{s}{a}}\operatorname{sh}\sqrt{\frac{s}{a}}\,R +T_k^3\operatorname{ch}\sqrt{\frac{s}{a}}\,R +3T_k^2(T_{c0}-T_k)\operatorname{ch}\sqrt{\frac{s+\kappa}{a}}\,R\\ &\quad +3T_k(T_{c0}-T_k)^2\operatorname{ch}\sqrt{\frac{s+2\kappa}{a}}\,R +(T_{c0}-T_k)^3\operatorname{ch}\sqrt{\frac{s+3\kappa}{a}}\,R \Bigg] \\ &= \frac{T_k^3(T_k-T_0)}{s} +\frac{T_k^2(T_{c0}-T_k)(4T_k-3T_0)}{s+\kappa} +\frac{3T_k(T_{c0}-T_k)^2(2T_k-T_0)}{s+2\kappa} \\ &\quad +\frac{(T_{c0}-T_k)^3(4T_k-T_0)}{s+3\kappa} +\frac{(T_{c0}-T_k)^4}{s+4\kappa}. \tag{I.2} \end{aligned} \]

Substituting the value of \(A\) from (I.2) into (I.1), we obtain the solution of the problem in transforms

\[ \overline{T}(x,s)-\frac{T_0}{s} = \Big[ T_k^3(T_k-T_0)(s+\kappa)(s+2\kappa)(s+3\kappa)(s+4\kappa) + \]

\[ +\,T_k^2(T_{c0}-T_k)(4T_k-3T_0)\,s(s+2\kappa)(s+3\kappa)(s+4\kappa) +T_k(T_{c0}-T_k)^2\cdot \]

\[ \begin{gathered} \cdot (6T_{\kappa}-3T_0)s(s+\kappa)(s+3\kappa)(s+4\kappa) +(T_{c0}-T_{\kappa})^3(4T_{\kappa}-T_0)\cdot{}\\ \cdot s(s+\kappa)(s+2\kappa)(s+4\kappa) +(T_{c0}-T_{\kappa})^4s(s+\kappa)(s+2\kappa)\cdot{}\\ \cdot(s+3\kappa) \biggr]\operatorname{ch}\sqrt{\frac{s}{a}}\,x\cdot \biggl\{s(s+\kappa)(s+2\kappa)(s+3\kappa)(s+{}\\ +4\kappa) \biggl[ \psi\sqrt{\frac{s}{a}}\operatorname{sh}\sqrt{\frac{s}{a}}\,R +T_{\kappa}^{3}\operatorname{ch}\sqrt{\frac{s}{a}}\,R+{}\\ +3T_{\kappa}^{2}(T_{c0}-T_{\kappa})\operatorname{ch}\sqrt{\frac{s+\kappa}{a}}\,R +3T_{\kappa}(T_{c0}-T_{\kappa})^{2}\operatorname{ch}\sqrt{\frac{s+2\kappa}{a}}\,R+{}\\ +(T_{c0}-T_{\kappa})^{3}\operatorname{ch}\sqrt{\frac{s+3\kappa}{a}}\,R \biggr\}^{-1}. \end{gathered} \tag{I.3} \]

Expression (I.3) satisfies all the requirements of the expansion theorem. The denominator of this expression has the roots

\[ s_1=0,\qquad s_2=-\kappa,\qquad s_3=-2\kappa,\qquad s_4=-3\kappa,\qquad s_5=-4\kappa \]

and the set of roots

\[ s_n=-\frac{a\mu_n^2}{R^2}, \]

where \(\mu_n\) are the roots of the transcendental equation

\[ \operatorname{ctg}\mu_n = -\frac{\mu_n}{4\mathrm{Ki}} - \frac{3(\Theta_{c0}-1)\cos\sqrt{\mu_n^2-\mathrm{Pd}}}{\sin\mu_n} - \]

\[ - \frac{3(\Theta_{c0}-1)^2\cos\sqrt{\mu_n^2-2\mathrm{Pd}}}{\sin\mu_n} - \frac{(\Theta_{c0}-1)^3\cos\sqrt{\mu_n^2-3\mathrm{Pd}}}{\sin\mu_n}. \tag{I.4} \]

Passing from images to the originals of the functions, we obtain the general solution of the problem

\[ T(x,\tau)=T_0+C_j\sum_{j=0}^{j=4} \left[ \frac{ \cos\sqrt{\frac{j\kappa}{a}}\,x\,\exp(-j\kappa\tau) }{ \displaystyle \sum_{p=0}^{p=3}C_p\cos\sqrt{\frac{(j-p)\kappa}{a}}\,R -\psi\sqrt{\frac{j\kappa}{a}}\sin\sqrt{\frac{j\kappa}{a}}\,R } - \right. \]

\[ \left. -\sum_{n=1}^{\infty} \frac{A_n}{\mu_n^2-\dfrac{j\kappa R^2}{a}} \cos\mu_n\frac{x}{R} \exp\left(-\mu_n^2\frac{a\tau}{R^2}\right) \right], \tag{I.5} \]

where

\[ A_n= \left[ \frac{\psi}{2R}\left(\cos\mu_n+\frac{\sin\mu_n}{\mu_n}\right) + \frac{1}{2}\sum_{p=0}^{p=3} C_p \frac{\sin\sqrt{\mu_n^2-p\,\mathrm{Pd}}}{\sqrt{\mu_n^2-p\,\mathrm{Pd}}} \right]^{-1}. \tag{I.6} \]

Introducing dimensionless complexes, we obtain the solution in criterion form

\[ \Theta\left(\frac{x}{R},\mathrm{Fo}\right) = \Theta_0+C_j^{*}\sum_{j=0}^{j=4} \left[ \frac{ \cos\sqrt{j\mathrm{Pd}}\dfrac{x}{R}\exp(-j\mathrm{Pd}\,\mathrm{Fo}) }{ \displaystyle \sum_{p=0}^{p=3}C_p^{*}\sqrt{(j-p)\mathrm{Pd}} - \frac{1}{4\mathrm{Ki}}\sqrt{j\mathrm{Pd}}\sin\sqrt{j\mathrm{Pd}} } \right. \]

\[ -\sum_{n=1}^{\infty}\frac{A_n^*}{\mu_n^2-j\,\mathrm{Pd}}\cos\mu_n\frac{x}{R}\exp\left(-\mu_n^2\mathrm{Fo}\right)\right]. \tag{I.7} \]

Here

\[ A_n^*=\left[\frac{1}{8\mathrm{Ki}}\left(\cos\mu_n+\frac{\sin\mu_n}{\mu_n}\right)+ \frac{1}{2}\sum_{p=0}^{3}C_p^*\frac{\sin\sqrt{\mu_n^2-p\,\mathrm{Pd}}}{\sqrt{\mu_n^2-p\,\mathrm{Pd}}}\right]^{-1}. \tag{I.8} \]

The values of the coefficients \(C_j,\ C_j^*,\ C_p,\ C_p^*\) are given in Tables 1 and 2.

Table 1

Expressions for the coefficients \(C_j\) and \(C_j^*\)

Table 2

Expressions for the coefficients \(C_p\) and \(C_p^*\)

II. Temperature Field of an Infinite Cylinder

The temperature distribution is found as the solution of differential equation (1) with boundary conditions of the type (2), (3), (7) for \(\xi=r,\ \xi=2\).

Transforming equations (1), (2), (3) into images, we obtain the solution

\[ \overline{T}(r,s)-\frac{T_0}{s} = A J_0\left(\sqrt{\frac{s}{a}}\,r\right). \tag{II.1} \]

To transform boundary condition (7) into images, we use the displacement theorem and from the latter find the constant \(A\).

Then the solution of the problem in images is

\[ \overline{T}(r,s)-\frac{T_0}{s} = \left[T_k^3(T_k-T_0)(s+k)(s+2k)(s+3k)(s+4k)+ \right. \]

\[ \left. +\,T_k^2(T_{c0}-T_k)(4T_k-3T_0)s(s+2k)(s+3k)(s+4k)+ \right. \]

\[ \begin{gathered} + T_k (T_{c0}-T_k)^2(6T_k-3T_0)s(s+\kappa)(s+3\kappa)(s+4\kappa)+\\ +(T_{c0}-T_k)^3(4T_k-T_0)s(s+\kappa)(s+2\kappa)(s+4\kappa)+\\ +(T_{c0}-T_k)^4s(s+\kappa)(s+2\kappa)(s+3\kappa) \end{gathered} \]
\[ \cdot J_0\left(\sqrt{\frac{s}{a}}\,r\right) \left\{s(s+\kappa)(s+2\kappa)(s+3\kappa)(s+4\kappa)\right. \]
\[ \cdot \left[ T_k^3J_0\left(\sqrt{\frac{s}{a}}\,R\right) +3T_k^2(T_{c0}-T_k)J_0\left(\sqrt{\frac{s+\kappa}{a}}\,R\right)+ \]
\[ +3T_k(T_{c0}-T_k)^2J_0\left(\sqrt{\frac{s+2\kappa}{a}}\,R\right) +(T_{c0}-T_k)^3J_0\left(\sqrt{\frac{s+3\kappa}{a}}\,R\right)+ \]
\[ \left. +\psi\sqrt{\frac{s}{a}}\,J_1\left(\sqrt{\frac{s}{a}}\,R\right) \right]\right\}^{-1}. \tag{II.2} \]

Both the numerator and the denominator are polynomials in \(s\), and the polynomial of the denominator is of higher degree than the polynomial of the numerator. Therefore the expansion theorem may be applied to relation (II.2). The denominator of this expression contains the roots \(s_1=0\), \(s_2=-\kappa\), \(s_3=-2\kappa\), \(s_4=-3\kappa\), \(s_5=-4\kappa\), and an infinite set of roots

\[ s_n=-\frac{a\psi_n^2}{R^2}, \]

where \(\psi_n\) is the root of the transcendental equation

\[ J_0(\psi_n)+3(\Theta_{c0}-1)J_0\left(\sqrt{\mathrm{Pd}-\psi_n^2}\right) +3(\Theta_{c0}-1)^2J_0\left(\sqrt{2\mathrm{Pd}-\psi_n^2}\right)+ \]
\[ +(\Theta_{c0}-1)^3J_0\left(\sqrt{3\mathrm{Pd}-\psi_n^2}\right) =\frac{1}{4\mathrm{Ki}}\psi_nJ_1(\psi_n). \tag{II.3} \]

Passing from the transform to the original gives the general solution of the problem

\[ T(r,\tau)=T_0+C_j\sum_{j=0}^{j=4} \left[ \frac{ J_0\left(\sqrt{\frac{j\kappa}{a}}\,r\right)\exp(-j\kappa\tau) }{ \displaystyle \sum_{p=0}^{p=3}C_pJ_0\left(\sqrt{\frac{(j-p)\kappa}{a}}\,R\right) -\psi\sqrt{\frac{j\kappa}{a}}\,J_1\left(\sqrt{\frac{j\kappa}{a}}\,R\right) } \right. \]
\[ \left. -\sum_{n=1}^{\infty} \frac{A_n}{ \psi_n^2-\dfrac{j\kappa R^2}{a} } J_0\left(\psi_n\frac{r}{R}\right) \exp\left(-\psi_n^2\frac{a\tau}{R^2}\right) \right]. \tag{II.4} \]

Here

\[ A_n= \left[ \frac{\psi}{2R}\left(J_0(\psi_n)+\frac{J_1(\psi_n)}{\psi_n}\right) +\frac{1}{2}\sum_{p=0}^{p=3}C_p \frac{ J_1\left(\sqrt{\psi_n^2-\dfrac{p\kappa R^2}{a}}\right) }{ \sqrt{\psi_n^2-\dfrac{p\kappa R^2}{a}} } \right]^{-1}. \tag{II.5} \]

Separating out dimensionless complexes, we write the solution in criterion form

\[ \Theta\left(\frac{r}{R},\mathrm{Fo}\right)=\Theta_0+C_j^* \sum_{j=0}^{j=4} \left[ \frac{ J_0\left(\sqrt{j\mathrm{Pd}}\,\dfrac{r}{R}\right)\exp(-j\mathrm{Pd}\,\mathrm{Fo}) }{ \displaystyle \sum_{j=0}^{p=3}C_p^*J_0\left(\sqrt{(j-p)\mathrm{Pd}}\right) -\frac{1}{4\mathrm{Ki}}\sqrt{j\mathrm{Pd}}\,J_1\left(\sqrt{j\mathrm{Pd}}\right) } - \]

\[ -\sum_{n=1}^{\infty}\frac{A_n^*}{\psi_n^2-j\mathrm{Pd}}\, J_0\!\left(\psi_n\frac{r}{R}\right)\exp(-\psi_n^2\mathrm{Fo}) \right]. \tag{II.6} \]

Here

\[ A_n^*=\left[ \frac{1}{8\mathrm{Ki}}\left(J_0(\psi_n)+\frac{J_1(\psi_n)}{\psi_n}\right) +\frac{1}{2}\sum_{p=0}^{p=3} C_p^* \frac{J_1\!\sqrt{\psi_n^2-p\mathrm{Pd}}}{\sqrt{\psi_n^2-p\mathrm{Pd}}} \right]^{-1}. \tag{II.7} \]

III Temperature field of a sphere

Solving equation (1) analogously for \(\zeta=r,\ \xi=3\) with conditions (2), 3), (7), we obtain the temperature distribution in the sphere

\[ T(r,\tau)=T_0+ \]

\[ +\,C_j\sum_{j=0}^{j=4}\left[ \frac{ \dfrac{\sin\!\left(\sqrt{\dfrac{j\kappa}{a}}\,r\right)}{\dfrac{r}{R}} \exp(-j\kappa\tau) }{ \displaystyle \sum_{p=0}^{p=3}C_p\sin\!\left(\sqrt{\frac{(j-p)\kappa}{a}}\,R\right) +\psi\sqrt{\frac{j\kappa}{a}}\cos\!\left(\sqrt{\frac{j\kappa}{a}}\,R\right) -\frac{\psi}{R}\sin\!\left(\sqrt{\frac{j\kappa}{a}}\,R\right) } - \sum_{n=1}^{\infty} \frac{A_n}{\psi_n^2-\dfrac{j\kappa R^2}{a}}\cdot \frac{\sin\psi_n\dfrac{r}{R}}{\dfrac{r}{R}} \exp\!\left(-\psi_n^2\frac{a\tau}{R^2}\right) \right]. \tag{III.1} \]

In this case

\[ A_n=\left[ \frac{\psi}{2R}\sin\psi_n -\frac{1}{2}\sum_{p=0}^{p=3}C_p \frac{ \cos\sqrt{\psi_n^2-\dfrac{p\kappa R^2}{a}} }{ \sqrt{\psi_n^2-\dfrac{p\kappa R^2}{a}} } \right]^{-1}, \tag{III.2} \]

and \(\psi_n\) is the root of the transcendental equation

\[ \tan\psi_n= \frac{\psi_n}{4\mathrm{Ki}-1} -\frac{3(\Theta_{c0}-1)}{1-\dfrac{1}{4\mathrm{Ki}}}\cdot \frac{\sin\sqrt{\mathrm{Pd}-\psi_n^2}}{\cos\psi_n} - \]

\[ -\frac{3(\Theta_{c0}-1)^2}{1-\dfrac{1}{4\mathrm{Ki}}}\cdot \frac{\sin\sqrt{2\mathrm{Pd}-\psi_n^2}}{\cos\psi_n} -\frac{(\Theta_{c0}-1)^3}{1-\dfrac{1}{4\mathrm{Ki}}}\cdot \frac{\sin\sqrt{3\mathrm{Pd}-\psi_n^2}}{\cos\psi_n}. \tag{III.3} \]

Passing to dimensionless form, we obtain

\[ \Theta\!\left(\frac{r}{R},\mathrm{Fo}\right)=\Theta_0+ \]

\[ +C_j^*\sum_{j=0}^{j=4}\left[ \frac{ \dfrac{\sin\sqrt{j\mathrm{Pd}}\,\dfrac{r}{R}}{\dfrac{r}{R}} \exp(-j\mathrm{Pd}\,\mathrm{Fo}) }{ \displaystyle \sum_{p=0}^{p=3}C_p^*\cos\sqrt{(j-p)\mathrm{Pd}} +\frac{1}{4\mathrm{Ki}}\left(\sqrt{j\mathrm{Pd}}\cos\sqrt{j\mathrm{Pd}}-\sin\sqrt{j\mathrm{Pd}}\right) } - \right. \]

\[ -\sum_{n=1}^{\infty}\frac{A_n^*}{\nu_n^2-j\mathrm{Pd}}\cdot \frac{\sin \nu_n \dfrac{r}{R}}{\dfrac{r}{R}}\, \exp(-\nu_n^2\mathrm{Fo}) \biggr]. \tag{III.4} \]

Here

\[ A_n^*= \left[ \frac{1}{8\mathrm{Ki}}\,\nu_n -\frac{1}{2}\sum_{p=0}^{p=3} C_p^* \frac{\cos\sqrt{\nu_n^2-p\mathrm{Pd}}}{\sqrt{\nu_n^2-p\mathrm{Pd}}} \right]^{-1}. \tag{III.5} \]

Let us consider an example of using solution (II.6) for the case of heating an infinite cylinder of diameter \(700\ \mathrm{mm}\)

\[ \left(\lambda=34.9\,\frac{\mathrm{W}}{\mathrm{m}\cdot \mathrm{deg}},\quad a=0.0225\,\mathrm{m}^2/\mathrm{h}\right) \]

in a furnace with variable temperature

\[ T_c(\tau)=1573-(1573-1353)\exp(-\tau)\ ^\circ K \]

and with the reduced radiation coefficient

\[ \sigma_v=3.49\,\frac{\mathrm{W}}{\mathrm{m}^2\mathrm{deg}^4}. \]

The initial temperature of the cylinder is \(T_0=873^\circ K\).
We first compute \(\Theta_0=0.555;\quad \Theta_{c0}=0.86;\quad \mathrm{Pd}=5.44;\quad \mathrm{Ki}=1.362.\)

The values of the roots of the characteristic equation (II.3) were found graphically. Figure 1 shows the determination of the first three roots of this equation, where it was assumed that

Fig. 1. Determination of the roots of the transcendental equation.

\[ y_1=\frac{J_0(\mu_n)}{J_1(\mu_n)}, \]

\[ y_2= \frac{\nu_n}{4\mathrm{Ki}} -3(\Theta_{c0}-1)\frac{J_0\!\left(\sqrt{\mathrm{Pd}-\mu_n^2}\right)}{J_1(\mu_n)} -3(\Theta_{c0}-1)^2 \frac{J_0\!\left(\sqrt{2\mathrm{Pd}-\mu_n^2}\right)} {\sqrt{2\mathrm{Pd}-\mu_n^2}} - \]

\[ - \left(\Theta_{co}-1\right)\, \frac{J_0\left(\sqrt{3\mathrm{Pd}-\mu_n^2}\right)} {\sqrt{3\mathrm{Pd}-\mu_n^2}} . \]

Under the specified conditions, \(\mu_1 = 1.71;\ \mu_2 = 4.38;\ \mu_3 = 7.42\).

Let us compare the calculation data with the experimental and computed curves presented in [2]. Figure 2 shows the curves of the change in the temperature of the radiating medium, the surface temperature, and the center temperature: the solid lines are based on experimental data, the dash-dotted lines are according to the calculation by (II.6), and the dashed lines are the computed data of V. N. Sokolov [2].

The results of the comparison showed that the proposed solutions describe, with accuracy sufficient for practice, the temperature distribution in solids when the temperatures of the medium and of the surface differ little from one another.

Fig. 2. Change in the temperature of the surface and center of a hot cylindrical ingot.

Analysis of the temperature fields makes it possible to recommend relations (I.7), (II.6), (III.4) for calculating the radiative heating of hot ingots when the ratio of the surface temperature to the temperature of the radiator at a given moment is greater than 0.65. In the case of a considerable excess of the medium temperature over the surface temperature, the computational equations of [3] may be used to calculate the temperature distribution.

Solutions for other laws of variation of the temperature of the radiating medium (linear, parabolic, etc.) are carried out in an analogous manner.

Principal notation

\(\zeta\) — generalized coordinate,

\(\xi\) — coefficient characterizing the shape of the body,

\(T_o\) — absolute initial temperature of the body,

\(T_{co}\) — absolute initial temperature of the medium,

\(T_k\) — maximum absolute temperature of the radiating medium,

\[ \mathrm{Pd}=\frac{\kappa R^{2}}{a} \]
—the Predvoditelev criterion,

\[ F_{o}=\frac{a\tau}{R^{2}} \]
—the Fourier criterion,

\[ \mathrm{Ki}=\frac{\varepsilon_{n} C_{o}\left(\dfrac{T_{k}}{100}\right)^{4} R}{\lambda T_{k}} \]
—the Kirpichev criterion,

\[ \Theta=\frac{T}{T_{k}} \]
—the dimensionless temperature,

\(J_{o}, J_{1}\)—Bessel functions of the first kind of orders zero and one.

REFERENCES

1. A. V. Lykov. Theory of Heat Conduction. GITTL, 1952.
2. V. I. Sokolov. Calculations of Metal Heating in Metallurgical Furnaces. Metallurgizdat, 1956.
3. V. V. Salomatov, G. P. Boikov. Izvestiya Vuzov. Ferrous Metallurgy, No. 12, 1963.