On the Temperature Field in a Plane Thick-Walled Shield

Yu. V. Vidin, G. P. Boikov

(Submitted by Prof. G. I. Fuks, Doctor of Technical Sciences)

Very often one has to deal with the need to determine the steady-state temperature field in a plane wall. Of great interest are examples in which heat exchange at the boundaries of the body occurs according to the Stefan–Boltzmann law.

In the literature, such problems are recommended to be solved by trial selection. It is known that, in a number of cases, the trial-selection process may prove very laborious and technically inconvenient. This is precisely the case when the boundary conditions are expressed by the difference of the fourth powers of absolute temperatures.

Below a method is proposed that makes it possible to apply the method of velocity iteration to determine the temperature field in a plane thick-walled shield.

For an unbounded plate, a problem of this kind can be formulated mathematically in the following form:

\[ \frac{d}{dx}\left[\lambda(T)\frac{dT(x)}{dx}\right]=0; \tag{1} \]

\[ \lambda(T)\big|_{T(+R)}\cdot \frac{dT(+R)}{dx} = \varepsilon_{n_1}\cdot C_0 \left[ \left(\frac{T_{c_1}}{100}\right)^4 - \left(\frac{T(+R)}{100}\right)^4 \right]; \tag{2} \]

\[ -\lambda(T)\big|_{T(-R)}\frac{dT(-R)}{dx} = \varepsilon_{n_2}\cdot C_0 \left[ \left(\frac{T_{c_2}}{100}\right)^4 - \left(\frac{T(-R)}{100}\right)^4 \right], \tag{3} \]

where \(T_{c_1}\) and \(T_{c_2}\) are the temperatures of the heating media for the left and right sides, respectively, \({}^{\circ}\mathrm{K}\);

\[ \varepsilon_{n_1},\ \varepsilon_{n_2} \]

are the values of the reduced degree of blackness, respectively for the left and right sides;

\(C_0\) is the radiation coefficient of an absolutely black body,

\[ C_0 = 5.7 \frac{\mathrm{W}}{\mathrm{m}^2\cdot{}^{\circ}\mathrm{K}^4}; \]

\(\lambda(T)\) is the thermal-conductivity coefficient of the body, depending in the general case on temperature,

\[ \frac{\mathrm{W}}{\mathrm{m}\cdot{}^{\circ}\mathrm{K}}. \]

\(T(+R), T(-R)\)—temperatures, respectively, of the left and right surfaces of the plate, °K;

\(2R\)—the total thickness of the plate, m.

The solution of the system of equations (1), (2), and (3) can be represented in the form

\[ \int \lambda(T)\,dT = \frac{\int \lambda(T)\cdot dT\big|_{T(-R)}-\int \lambda(T)\cdot dT\big|_{T(-R)}}{2}\,\frac{x}{R} + \frac{\int \lambda(T)\cdot dT\big|_{T(+R)}+\int \lambda(T)\cdot dT\big|_{T(-R)}}{2}. \tag{4} \]

To obtain this dependence in explicit form, it is necessary to know the dependence of the coefficient of thermal conductivity on temperature and the values of the temperatures of the body surfaces \(T(+R)\) and \(T(-R)\). These temperatures are determined on the basis of the prescribed boundary conditions (2) and (3). To find \(T(+R)\) and \(T(-R)\), two equations are obtained:

\[ \varepsilon_{n_1}\cdot C_0\cdot \left[ \left(\frac{T_{c_1}}{100}\right)^4 - \left(\frac{T(+R)}{100}\right)^4 \right] = -\varepsilon_{n_2}\cdot C_0 \left[ \left(\frac{T_{c_2}}{100}\right)^4 - \left(\frac{T(-R)}{100}\right)^4 \right]; \tag{5} \]

\[ \frac{ \int \lambda(T)\cdot dT\big|_{T(+R)} - \int \lambda(T)\cdot dT\big|_{T(-R)} }{2R} = \varepsilon_{n_1}\cdot C_0 \left[ \left(\frac{T_{c_1}}{100}\right)^4 - \left(\frac{T(+R)}{100}\right)^4 \right]. \tag{6} \]

Knowing \(\lambda=\lambda(T)\), from expression (6) we find \(T(-R)=f[T(+R)]\). Substitution of this relation into (5) gives

\[ \frac{\varepsilon_{n_1}}{\varepsilon_{n_2}} \left(\frac{T_{c_1}}{100}\right)^4 + \left(\frac{T_{c_2}}{100}\right)^4 = \frac{\varepsilon_{n_1}}{\varepsilon_{n_2}} \left(\frac{T(+R)}{100}\right)^4 + \left(\frac{f[T(+R)]}{100}\right)^4 . \tag{7} \]

As a rule, the degree of equation (7) is higher than four, and therefore the solution cannot be represented by means of radicals through its coefficients. Analysis of this expression for the number of real positive roots may be carried out on the basis of existing methods of higher algebra (Sturm’s method, Descartes’ rule, etc.). From physical considerations concerning the process, it is natural to expect the existence of only one real positive root of equation (7) of multiplicity greater than one. Finding this root is the task in solving relation (7).

This question may be clarified in the following way. Let us represent the second term of expression (7) in the form

\[ \left(\frac{f[T(+R)]}{100}\right)^4 = m^4\left(\frac{T(+R)}{100}\right)^4, \tag{8} \]

where the positive value \(m\) is equal to

\[ m=\frac{f[T(+R)]}{T(+R)}=\frac{T(-R)}{T(+R)}. \tag{9} \]

With this substitution, relation (7) takes the form

\[ \frac{\varepsilon_{n_1}}{\varepsilon_{n_2}} \left(\frac{T_{c_1}}{100}\right)^4 + \left(\frac{T_{c_2}}{100}\right)^4 = \left(\frac{T(+R)}{100}\right)^4 \cdot \left( \frac{\varepsilon_{n_1}}{\varepsilon_{n_2}}+m^4 \right). \tag{10} \]

Hence

\[ T(+R)=T_{c_1}\sqrt[4]{\frac{\dfrac{\varepsilon_{n_1}}{\varepsilon_{n_2}}+h^4}{\dfrac{\varepsilon_{n_1}}{\varepsilon_{n_2}}+m^4}}, \tag{11} \]

where \(h=\dfrac{T_{c_2}}{T_{c_1}}\ll 1\). It is conventionally assumed that \(T_{c_1}\gg T_{c_2}\) (the positive direction of the \(X\) axis is chosen toward the medium with the higher temperature).

In the case \(\varepsilon_{n_1}=\varepsilon_{n_2}\), we have

\[ T(+R)=T_{c_1}\sqrt[4]{\frac{1+h^4}{1+m^4}}. \tag{11′} \]

The bounds of the coefficient \(m\) can be established on the basis of examining (9) and (11). According to (9), the ratio \(m\) must not exceed unity, and from (11) it follows that the quantity \(m\) must be equal to or greater than the complex \(h\). Consequently, we have

\[ h\leq m\leq 1. \tag{12} \]

Substitution of these limiting values of \(m\) into solution (11) gives the upper and lower bounds for the actual value of the temperature \(T(+R)\), namely

\[ T_{c_1}\sqrt[4]{\frac{\dfrac{\varepsilon_{n_1}}{\varepsilon_{n_2}}+h^4}{\dfrac{\varepsilon_{n_1}}{\varepsilon_{n_2}}+1}} \leq T(+R)\leq T_{c_1}. \tag{12′} \]

When \(\varepsilon_{n_1}=\varepsilon_{n_2}\), then

\[ T_{c_1}\sqrt[4]{\frac{1+h^4}{2}}\leq T(+R)\leq T_{c_1}. \tag{12″} \]

This interval is sufficiently narrow, which makes it possible to reduce considerably the number of iterative operations.

Thus, proceeding from a preliminary body of knowledge, the value of the ratio \(m\), already in the first approximation, can be chosen sufficiently close to the true one if inequality (12) is used. Let initially \(m=m_1\). Then from (11) the first expression for the temperature of the plate surface \(T_1(+R)\) is found. After this, on the basis of (9), the adopted value \(m_1\) is refined. When

\[ m'_1=\frac{f[T_1(+R)]}{T_1(+R)} \]

differs substantially from \(m_1\), the calculation is then continued according to

\[ m_2=\frac{m_1+M_1}{2}, \]

where \(M_1=m'_1\), if \(m'_1>h\), and \(M_1=h\) in the case \(m'_1\leq h\); i.e., the establishment of the bounds of \(m\) for each subsequent operation is carried out with allowance for the limiting values of \(m\) in the preceding operation (see Table 1). The described process is repeated until the value \(m_n\) becomes equal to \(m'_n\). The temperature \(T_n(+R)\), determined according to (11) for \(m=m_n\), will be the desired quantity. Then

\[ T(-R)=m_n\cdot T_n(+R). \]

Let us consider the particular case of problems (1), (2), (3), when \(\lambda=\mathrm{const}\). Under this condition, expressions (4), (6), (7), and (9) take the form:

\[ T(x)=\frac{T(+R)-T(-R)}{2}\cdot\frac{x}{R}+\frac{T(+R)+T(-R)}{2}; \tag{4'} \]

\[ \frac{\lambda}{2R}\,[T(+R)-T(-R)] = \varepsilon_{n_1}\cdot C_0 \left[ \left(\frac{T_{c_1}}{100}\right)^4 - \left(\frac{T(+R)}{100}\right)^4 \right]; \tag{6'} \]

\[ \frac{\varepsilon_{n_1}}{\varepsilon_{n_2}} \left(\frac{T_{c_1}}{100}\right)^4 + \left(\frac{T_{c_2}}{100}\right)^4 = \frac{\varepsilon_{n_1}}{\varepsilon_{n_2}} \left(\frac{T(+R)}{100}\right)^4 + \left\{ \frac{ T(+R)-\dfrac{2R}{\lambda}\varepsilon_{n_1}\cdot C_0 \left[ \left(\dfrac{T_{c_1}}{100}\right)^4 - \left(\dfrac{T(+R)}{100}\right)^4 \right] }{100} \right\}^4; \tag{7'} \]

\[ m=\frac{T(-R)}{T(+R)} = \sqrt[4]{ \left\{ 1- \frac{ \dfrac{2R}{\lambda}\varepsilon_{n_1}\cdot C_0\cdot \left[ \left(\dfrac{T_{c_1}}{100}\right)^4 - \left(\dfrac{T(+R)}{100}\right)^4 \right] }{T(+R)} \right\}^4 }. \tag{9'} \]

It is not difficult to see that relation \((7')\) is an algebraic equation of the sixteenth degree, the solution of which cannot be represented in radicals. From the fundamental theorem of algebra of complex numbers it follows [1] that this equation must have 16 roots. As has already been indicated, there are several methods for finding the exact number of real roots. Among them the most convenient is Sturm’s method [1]. In the case under consideration, however, this question can be resolved more simply if one makes use of the considerations set forth above.

From expression \((9')\) one can determine four values of the coefficient \(m\): one positive, one negative, and two complex. To these four values of the quantity \(m\), on the basis of relation (11), there correspond one positive and one negative root, each of multiplicity four. The other eight roots are complex.

From the physical point of view, the greatest interest is presented by the real, positive solution of dependence \((7')\), which can also be determined on the basis of \((9')\) and (11) by the method of successive approximation to the true value.

Table 1

Table 1 gives an example of a calculation \((T_{c_1}=1700^\circ K;\ T_{c_2}=1190^\circ K;\ h=0.7;\ \frac{\varepsilon_{n_1}}{\varepsilon_{n_2}}=0.753;\ \varepsilon_{n_2}=0.5;\ 2R=0.8\ \text{m};\ \lambda=51.15\ \text{W}/\text{m}\cdot{}^\circ K)\).

It is seen from this table that already at the second iteration the discrepancy between the calculated temperature and the actual one does not exceed the accuracy requirements of engineering calculation.

Figure 1 (solid line) shows the stationary temperature field obtained taking into account the data of Table 1. Here also (dashed lines) the process is depicted by which the nonstationary heating regime approaches its limiting state, calculated by the numerical method [2], under exactly the same external conditions.

Fig. 1. The process by which the nonstationary temperature field approaches the established thermal regime.

For the case when \(2R \to 0\) (or \(\lambda \to \infty\)), the calculated relations \((9')\) and (11) are transformed into the expressions:

\[ m=\frac{T(-R)}{T(+R)}=1; \tag{9''} \]

\[ T(+R)=T(-R)=T_{c_1}\sqrt[4]{\frac{\dfrac{\varepsilon_{n_1}}{\varepsilon_{n_2}}+h^4}{\dfrac{\varepsilon_{n_1}}{\varepsilon_{n_2}}+1}}, \tag{11''} \]

which are valid for a plane thin-walled shield.

REFERENCES

1. Kurosh A. G. Course of Higher Algebra. Fizmatgiz, 1952.
2. Vanichev A. P. Proceedings of NII-1. No. 25, 1947.