Relationship Between Excess Temperatures of a Body of Finite Dimensions
Abstract Generated abstract
This paper derives relations for estimating two-dimensional excess temperature fields in finite bodies from temperatures measured along symmetry axes or selected boundary lines. Starting from the heat-conduction equation, the authors assume a constant ratio between second derivatives in orthogonal directions and use a Fourier-form solution to obtain product formulas linking interior temperatures to axial, central, or edge temperatures. The proposed relations are checked against known unsteady solutions and numerical calculations for steady isotropic and anisotropic heat-transfer problems with internal heat generation. The results suggest that, for rectangular bars, parallelepipeds, and short cylinders, full temperature fields and inaccessible central temperatures can often be inferred from limited temperature measurements.
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Relationship Between Excess Temperatures of a Body of Finite Dimensions
G. I. Boikov, Yu. A. Korolenko
(Presented by Professor, Doctor I. D. Kutyavin)
A large number of heat-conduction phenomena can, in general form, be described by the differential equation
\[ \frac{\partial^2 \tau}{\partial x^2}+\frac{\partial^2 \tau}{\partial y^2}=F, \tag{1} \]
where \(F\) may depend on time (a nonstationary process) and may be independent of time (a stationary process); \(\tau=T-T_c\) is the excess temperature of the body (the difference between the temperature under consideration and the temperature of the surrounding medium).
We shall represent the original equation (1) in the form of two relations:
\[ \frac{\partial^2 \tau}{\partial x^2} \left( 1+\frac{\dfrac{\partial^2 \tau}{\partial y^2}} {\dfrac{\partial^2 \tau}{\partial x^2}} \right)=F; \qquad \frac{\partial^2 \tau}{\partial y^2} \left( 1+\frac{1} {\dfrac{\partial^2 \tau}{\partial y^2}\big/ \dfrac{\partial^2 \tau}{\partial x^2}} \right)=F. \]
Taking the ratio of the components of the divergence of the temperature gradient to be
\[ \frac{\dfrac{\partial^2 \tau}{\partial y^2}} {\dfrac{\partial^2 \tau}{\partial x^2}} =\xi=\mathrm{const}, \tag{1} \]
we find the dependence
\[ \frac{\partial^2 \tau}{\partial x^2} = \frac{1}{\xi}\cdot \frac{\partial^2 \tau}{\partial y^2}. \]
Solving the latter differential equation by the classical Fourier method, taking symmetry into account (the origin of coordinates at the center of the body), we obtain
\[ \tau(x;y)=D\cos(k\sqrt{\xi}\,y)\cos kx, \]
hence
\[ \vartheta(0;y)=D\cdot \cos k\sqrt{\zeta}\,y;\qquad \cos k\sqrt{\zeta}\,y=\frac{\vartheta(0;y)}{D}; \]
\[ \vartheta(x;0)=D\cdot \cos kx;\qquad \cos kx=\frac{\vartheta(x;0)}{D}. \]
Consequently,
\[ \vartheta(x;y)=\frac{\vartheta(x;0)\cdot \vartheta(0;y)}{D}; \]
for \(x=y=0\) the temperature at the center of an infinite rectangular bar must be obtained. This gives grounds for asserting that \(D=\vartheta(0;0)\) is the temperature of the center.
Therefore
\[ \vartheta(x;y)=\frac{\vartheta(x;0)\cdot \vartheta(0;y)}{\vartheta(0;0)}, \tag{2a} \]
in exactly the same way,
\[ \vartheta(R_1;y)=D\cdot \cos k\sqrt{\zeta}\,y\cdot \cos kR_1; \]
\[ \vartheta(x;R_2)=D\cdot \cos k\sqrt{\zeta}\,R_2\cdot \cos kx; \]
\[ \vartheta(x;y)=\frac{\vartheta(x;R_2)\cdot \vartheta(R_1;y)} {D\cdot \cos kR_1\cdot \cos k\sqrt{\zeta}\,R_2}; \]
for \(x=R_1;\ y=R_2\), the temperature of the edge of an infinite rectangular bar must be obtained. This gives grounds for asserting that
\[ D\cdot \cos kR_1\cdot \cos k\sqrt{\zeta}\,R_2=\vartheta(R_1R_2); \]
therefore
\[ \vartheta(x;y)=\frac{\vartheta(x;R_2)\cdot \vartheta(R_1;y)} {\vartheta(R_1;R_2)}. \tag{2b} \]
For boundary conditions characterized by an infinite heat-transfer coefficient, formulas (2) give an indeterminacy of the form \(0\cdot 0\) and
\[ \frac{0\cdot 0}{0}. \]
The indeterminacy can be avoided only under the condition: 1) in formula (2a), do not seek the value \(\vartheta(R_1;R_2)\); 2) in formula (2b), instead of \(R_1\) and \(R_2\), take somewhat smaller values (for example, \(0.9R_1;\ 0.9R_2\) or \(0.8R_1;\ 0.8R_2\)). Since the case of boundary conditions with an infinite heat-transfer coefficient is very rare, and is for the most part created artificially, the reservation made is, in essence, of no practical significance. Formulas analogous to (2) for a parallelepiped and a cylinder of finite dimensions are obtained by an analogous method.
We shall demonstrate the validity of the noted regularity by a number of examples.
Unsteady process of heat transfer in an isotropic body
under boundary conditions of the first and third kind
According to [2]
\[ \vartheta(x;y)=\vartheta_0\cdot \sum_{n=1}^{\infty} A_n(z)\cdot \cos\mu_n\,\frac{x}{R_1}\cdot \sum_{m=1}^{\infty} A_m(z)\cdot \cos\mu_m\,\frac{y}{R_2}. \tag{3} \]
\[ \vartheta(0;y)=\vartheta_{0}\cdot \sum_{n=1}^{\infty} A_n(z)\sum_{m=1}^{\infty} A_m(z)\cdot \cos \psi_m \frac{y}{R_2}; \tag{б} \]
\[ \vartheta(x;0)=\vartheta_{0}\cdot \sum_{n=1}^{\infty} A_n(z)\cdot \cos \psi_n \frac{x}{R_1}\cdot \sum_{m=1}^{\infty} A_m(z); \tag{в} \]
\[ \vartheta(0;0)=\vartheta_{0}\cdot \sum_{n=1}^{\infty} A_n(z)\cdot \sum_{m=1}^{\infty} A_m(z). \tag{г} \]
Substituting (б), (в), (г) into (2а), after reduction we obtain the solution (а).
Steady-state process of heat transfer in an isotropic body under boundary conditions of the third kind and with internal heat generation
For a bar of rectangular cross section \(2R_1=0.5\,\text{m}\); \(2R_2=1\,\text{m}\), with
\[ \lambda=40\,\frac{\text{kcal}}{\text{m}\cdot\text{h}\cdot\text{deg}}; \quad \alpha=30\,\frac{\text{kcal}}{\text{m}^2\cdot\text{h}\cdot\text{deg}}; \quad t_c=0^\circ\text{C}; \quad W=20000\,\frac{\text{kcal}}{\text{m}^3\cdot\text{h}} \]
the calculation of the temperature field by the method of elementary balances gave the results shown in Fig. 1 (the number in the lower left part of the cell); the results of calculations by formula (26) are also given there (the number in the upper right part of the cell). In view of symmetry, one quarter of the cross section of the bar is shown.
The system of equations of the elementary balances was solved on an electrical computer at TSU.
In Fig. 1 point 1 is the center of the body. The \(x\)-axis goes from point 1, passing through points 2 and 3. The \(y\)-axis goes from 1 through 4 and 7. The temperatures in the cells refer to the point at the middle of the cell. Calculations by formula (26) were carried out as follows:
\[ \vartheta_1=\frac{\vartheta_3\cdot \vartheta_7}{\vartheta_0}=131 \qquad \left( \text{since } \vartheta=t, \ t_c=0 \right) \]
\[ \vartheta_5=\frac{\vartheta_8\cdot \vartheta_6}{\vartheta_0}=126.5 \quad \text{and so on:} \quad \vartheta_4=128;\ \vartheta_2=129. \]
Fig. 1.
Steady-state heat transfer in an anisotropic body under boundary conditions of the third kind with internal heat generation
Figure 2 gives data for the temperature field in the form \(\vartheta=t-35^\circ\text{C}\), obtained by numerical solution of the differential equation of heat conduction in the cross section of the magnet core of a betatron [3] (the number
in the left lower part of the cell). Here calculations by formula (26) are also given (the number in the upper right part of the cell). In view of the symmetry, only half of the section is shown.
Fig. 2.
The calculations by formula (26) were carried out as follows:
\[ \vartheta_1=\frac{\vartheta_{13}\cdot \vartheta_4}{\vartheta_{16}}=63.6; \qquad \vartheta_{10}=\frac{\vartheta_{14}\cdot \vartheta_{12}}{\vartheta_{16}}=54.5 \]
and so on: \(\vartheta_6=57.6;\ \vartheta_9=58.3.\)
Thus, in investigating the temperature field in bodies (when heat propagates in more than one dimension), it is sufficient to find the distribution of temperatures along the axes of symmetry of the body (a bar of rectangular cross section, a parallelepiped, a short cylinder). The temperature values for the remaining part of the field can be calculated indirectly, by formula (2a). Formula (26) is of particular value for the same purpose. It makes it possible to calculate accurately the temperature field of the entire mass of the body if the corresponding temperatures on the surface are accurately measured (for a bar of rectangular cross section, a parallelepiped, and a cylinder of finite dimensions). It makes it possible to determine the temperature at the center of the body when, because of technological conditions, it is impossible to reach it with a thermocouple.
LITERATURE
- G. P. Boikov. Heating of bodies of finite dimensions under the action of radiant heat. Izv. TPI, vol. 101, 1958.
- A. V. Lykov. Theory of Heat Conduction, GITTL, Moscow, 1952.
- A. A. Gurchenok. Calculation of heat transfer in cooling plates of the magnetic circuit of air-cooled betatrons. Izv. Vyssh. Shkoly. Elektromekhanika, No. 2, 1959.