Stationary Temperature Fields in Some Bodies of Revolution

V. V. Ivanov

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

Spherical Segment

Statement of the problem: find the stationary distribution of temperature in a spherical segment of radius \(R\), if the surfaces of the sphere and of the base are maintained respectively at temperatures \(U\) and zero (Fig. 1a).

Let us first determine the temperature field in a sphere of radius \(R\), when the part of its surface corresponding to the given spherical segment remains at temperature \(U\), while the other part has temperature \((-U)\) (Fig. 1b). The problem reduces to the integration of Laplace’s equation:

\[ \frac{\partial}{\partial r}\left(r^2\frac{\partial T}{\partial r}\right) + \frac{1}{\sin\Theta}\cdot \frac{\partial}{\partial\Theta} \left(\sin\Theta\,\frac{\partial T}{\partial\Theta}\right) =0 \]

with boundary conditions

\[ T(r,\Theta)_{r=R}= \begin{cases} U, & \text{for } 0\le \Theta<\beta,\\ -U, & \text{for } \beta<\Theta\le \pi. \end{cases} \]

a  Fig. 1.  b

The solution is constructed in the form of the series

\[ T(r,\Theta)=\sum_{n=0}^{\infty} a_n \left(\frac{r}{R}\right)^n P_n(\cos\Theta), \]

where \(P_n(x)\) are Legendre polynomials of order \(n\). The constants \(a_n\) are determined from the boundary conditions

\[ a_n=\frac{2n+1}{2}\int T(R,\Theta)P_n(\cos\Theta)\,d(\cos\Theta). \]

Or, putting \(\cos\Theta=x\),

\[ a_n=\frac{2n+1}{2}\left[\int_{-1}^{\cos\beta}(-U)P_n(x)\,dx+\int_{\cos\beta}^{1}UP_n(x)\,dx\right]. \]

Evaluating the integrals by means of the recurrence formula

\[ (2n+1)P_n(x)=P'_{n+1}(x)-P'_{n-1}(x), \]

we finally find

\[ T(r,\Theta)=U\left\{-\cos\beta+\sum_{n=1}^{\infty}\left[P_{n-1}(\cos\beta)-P_{n+1}(\cos\beta)\right]\left(\frac{r}{R}\right)^n P_n(\cos\Theta)\right\}. \tag{1} \]

Isotherms inside the sphere have the form of spherical surfaces ending on the circumference \(MM\), except for one isotherm—a plane passing through the point \(A(R\cos\beta,\pi)\) and serving as the base of the given segment (Fig. 1b), for which

\[ T_{\mathrm{base}}=T(R\cos\beta,\pi)=U\left\{-\cos\beta+\sum_{n=1}^{\infty}\left[P_{n-1}(\cos\beta)-\right.\right. \]

\[ \left.\left.-P_{n+1}(\cos\beta)\right](\cos\beta)^n(-1)^n\right\}. \tag{2} \]

Subtracting (2) from (1), we obtain the temperature distribution in the spherical segment whose base is maintained at \(0^\circ\), while the spherical surface is maintained at the temperature:

\[ U-T_{\mathrm{base}}=U\left\{1+\cos\beta-\sum_{n=1}^{\infty}\left[P_{n-1}(\cos\beta)-P_{n+1}(\cos\beta)\right](\cos\beta)^n(-1)^n\right\}. \tag{3} \]

Now dividing the difference \(T(r,\Theta)-T_{\mathrm{base}}\) by the expression in braces in formula (3), we find the required temperature field

\[ t(r,\Theta)=U\, \frac{ \sum_{n=1}^{\infty}\left[P_{n-1}(\cos\beta)-P_{n+1}(\cos\beta)\right] \left[\left(\frac{r}{R}\right)^n P_n(\cos\Theta)-(\cos\beta)^n(-1)^n\right] }{ 1+\cos\beta-\sum_{n=1}^{\infty}\left[P_{n-1}(\cos\beta)-P_{n+1}(\cos\beta)\right](\cos\beta)^n(-1)^n }. \tag{4} \]

If the angle \(\beta\) is acute, then in equation (4) the factor \((-1)^n\) drops out. Setting \(\cos\beta=0\) and using the relations

\[ P_{n-1}(0)-P_{n+1}(0)=P_{n-1}(0)\,\frac{2n+1}{n+1}, \]

\[ P_{2n}(0)=(-1)^n\,\frac{1\cdot3\cdot5\cdots(2n-1)}{n!\,2^n} \quad\text{and}\quad P_{2n+1}(0)=0, \]

we obtain, as a special case of formula (4), the stationary temperature distribution in a hemisphere:

\[ t(r,\Theta)=U\sum_{n=0}^{\infty}(-1)^n \frac{1\cdot3\cdot5\cdots(2n-1)(4n+3)} {2\cdot4\cdot6\cdots2n(2n+2)} \left(\frac{r}{R}\right)^{2n+1} P_{2n+1}(\cos\Theta), \tag{5} \]

which corresponds to the solution given in [3].

To find the heat flux through a spherical segment whose temperature field is given by equation (4), let us single out in it an isothermal surface—a hemisphere of radius \(R_0\) (Fig. 1a). In the stationary state, the quantity of heat \(Q_\beta\) passing through the surface of the spherical segment and through the selected surface of the hemisphere is the same.

If the temperature of the hemisphere is \(U_0\), then the heat through it is

\[ Q_\beta=\lambda\int_{0}^{\pi/2} \left(\frac{\partial t}{\partial r}\right)_{r=R_0} \cdot 2\pi R_0^2\sin\Theta\,d\Theta \left[\frac{\text{kcal}}{\text{hour}}\right]. \]

Finding the gradient from equation (5), we obtain

\[ Q_\beta=2\pi\lambda U_0 R_0 \sum_{n=0}^{\infty} \frac{(4n+3)[(2n)!]^2(2n+1)} {(2n+2)^2\,2^{4n}(n!)^4} \simeq \pi\lambda U_0 R_0\cdot 7.53. \]

Since the isotherm \(U_0\) passes through point \(B\) (Fig. 1a), substituting its coordinates \(r=R|\sin\beta+\cos\beta|\), \(\Theta=0\) into (4) and assigning various values \(\beta>\frac{\pi}{2}\), we find the dependence \(U_0=UF(\beta)\) for obtuse angles \(\beta\).

On the other hand, the isotherm \(U_1\), which is the surface of a spherical segment with angle \(\beta_1<\frac{\pi}{2}\), passes through point \(C\) (Fig. 1a).

Substituting the coordinates of point \(C\),

\[ r=R\left(\sin\beta\tan\frac{\beta_1}{2}+\cos\beta\right), \quad \Theta=0, \]

for a given constant \(\beta>\frac{\pi}{2}\) into equation (4), and replacing in it \(U=U_0/F(\beta)\), we find \(F(\beta_1)\)—the relation between \(U_1\) and \(U_0\) for acute angles \(\beta\). Finally:

\[ Q=7.53\,\pi\lambda R_0\,UF(\beta) \left[\frac{\text{kcal}}{\text{hour}}\right], \]

where \(R_0=R\sin\beta\) is the radius of the base of the spherical segment, and \(F(\beta)\) is found from the curve (Fig. 2).

Fig. 2.

Truncated prolate spheroid

Problem statement: find the stationary temperature distribution inside a truncated prolate spheroid (Fig. 3a), if the surface of revolution is maintained at temperature \(U\), and the cutting plane has temperature equal to zero.

Fig. 3.

The problem is solved analogously to the preceding one. First let us determine the temperature field in a prolate spheroid, if the part of its surface corresponding to the given truncated prolate spheroid remains at temperature \(U\), while the other part has temperature \((-U)\). We introduce degenerate ellipsoidal coordinates: \(\alpha,\beta,\varphi\) [1, 2]. Then the solution of Laplace’s equation

\[ \frac{1}{\operatorname{sh}\alpha}\,\frac{\partial}{\partial \alpha} \left[\operatorname{sh}\alpha\,\frac{\partial T}{\partial \alpha}\right] + \frac{1}{\sin\beta}\,\frac{\partial}{\partial \beta} \left[\sin\beta\,\frac{\partial T}{\partial \beta}\right] =0 \]

has the form

\[ T(\alpha,\beta)=\sum_{n=0}^{\infty} f_n \frac{P_n(\operatorname{ch}\alpha)}{P_n(\operatorname{ch}\alpha_0)} P_n(\cos\beta). \tag{6} \]

Here \(\alpha=\alpha_0\) is the equation of the surface of the given prolate spheroid, and \(f_n\) are constants determined from the boundary conditions

\[ f_n=\frac{2n+1}{2}\int_{-1}^{0} T(\alpha_0,\beta)P_n(\cos\beta)\,d(\cos\beta), \]

where

\[ T(\alpha,\beta)_{\alpha=\alpha_0}= \begin{cases} U, & \text{for } 0\le \beta<\beta_0,\\ -U, & \text{for } \beta_0<\beta\le \pi . \end{cases} \]

After substituting \(f_n\) into (6), we obtain

\[ T(\alpha,\beta)= U\left\{ -\cos\beta_0+ \sum_{n=1}^{\infty} \left[ P_{n-1}(\cos\beta_0)- P_{n+1}(\cos\beta_0) \right] \frac{P_n(\operatorname{ch}\alpha)}{P_n(\operatorname{ch}\alpha_0)} P_n(\cos\beta) \right\}. \tag{7} \]

One of the isotherms inside the prolate spheroid is a plane passing through the point \(A\) with coordinates \(\alpha=0\), \(\beta=\beta_A\), or \(\alpha=\alpha_A\), \(\beta=0\), and is the base for the given truncated prolate spheroid. Therefore

\[ T_{\mathrm{axis}}=T(0,\beta_A)= U\left\{ -\cos\beta_0+ \sum_{n=1}^{\infty} \left[ P_{n-1}(\cos\beta_0)- P_{n+1}(\cos\beta_0) \right] \frac{P_n(\cos\beta_A)}{P_n(\operatorname{ch}\alpha_0)} \right\}. \tag{8} \]

Subtracting (8) from (7) and dividing this difference by the factor in braces in the expression

\[ U-T_{\mathrm{axis}}= U\left\{ 1+\cos\beta_0- \sum_{n=1}^{\infty} \left[ P_{n-1}(\cos\beta_0)- P_{n+1}(\cos\beta_0) \right] \frac{P_n(\cos\beta_A)}{P_n(\operatorname{ch}\alpha_0)} \right\}, \]

we obtain the desired temperature field:

\[ t(\alpha,\beta)= U\, \frac{ \displaystyle \sum_{n=1}^{\infty} \left[ P_{n-1}(\cos\beta_0)-P_{n+1}(\cos\beta_0) \right] \left[ \frac{ P_n(\operatorname{ch}\alpha)P_n(\cos\beta)-P_n(\cos\beta_A) }{ P_n(\operatorname{ch}\alpha_0) } \right] }{ \displaystyle 1+\cos\beta_0- \sum_{n=1}^{\infty} \left[ P_{n-1}(\cos\beta_0)-P_{n+1}(\cos\beta_0) \right] \frac{P_n(\cos\beta_A)}{P_n(\operatorname{ch}\alpha_0)} }, \]

where \(\cos\beta_A=\operatorname{ch}\alpha_0\cos\beta_0\).

When the cutting plane passes through the point \(A(\alpha_A,0)\), the solution has the form

\[ t(\alpha,\beta)= \]

\[ = U \frac{ \displaystyle \sum_{n=1}^{\infty} \bigl[P_{n-1}(\cos\beta_0)-P_{n+1}(\cos\beta_0)\bigr] \left[ \frac{ P_n(\operatorname{ch}\alpha)\cdot P_n(\cos\beta)-P_n(\operatorname{ch}\alpha_A) }{ P_n(\operatorname{ch}\alpha_0) } \right] }{ \displaystyle 1+\cos\beta_0- \sum_{n=1}^{\infty} \bigl[P_{n-1}(\cos\beta_0)-P_{n+1}(\cos\beta_0)\bigr] \frac{P_n(\operatorname{ch}\alpha_A)}{P_n(\operatorname{ch}\alpha_0)} }. \]

where \(\operatorname{ch}\alpha_A=\operatorname{ch}\alpha_0\cos\beta_0\).

For the case of a half prolate spheroid \((\alpha_A=0,\ \beta_A=\pi/2,\ \beta_0=\pi/2)\),

\[ t(\alpha,\beta)= U\sum_{n=0}^{\infty}(-1)^n \frac{1\cdot3\cdot5\cdots(2n-1)(4n+3)} {2\cdot4\cdot6\cdots(2n+2)} \frac{P_{2n+1}(\operatorname{ch}\alpha)} {P_{2n+1}(\operatorname{ch}\alpha_0)} P_{2n+1}(\cos\beta), \]

which corresponds to the solution given in [4].

Truncated oblate spheroid

Let us determine the temperature field in a truncated oblate spheroid, if the temperatures of the cutting plane and of the surface of revolution are equal respectively to zero and \(U\) (Fig. 3b). Using the expressed ellipsoidal coordinates \(\alpha,\beta,\varphi\) [1, 2] and without repeating the arguments of the preceding sections, we find that the solution of the Laplace equation

\[ \frac{1}{\operatorname{ch}\alpha}\frac{\partial}{\partial\alpha} \left(\operatorname{ch}\alpha\,\frac{\partial T}{\partial\alpha}\right) + \frac{1}{\sin\beta}\cdot\frac{\partial}{\partial\beta} \left(\sin\beta\cdot\frac{\partial T}{\partial\beta}\right)=0 \]

for the oblate spheroid, \(\alpha=\alpha_0\), has the form

\[ T(\alpha,\beta)= U\left\{ -\cos\beta_0+ \sum_{n=1}^{\infty} \bigl[P_{n-1}(\cos\beta_0)- P_{n+1}(\cos\beta_0)\bigr] \frac{P_n(i\,\operatorname{sh}\alpha)} {P_n(i\,\operatorname{sh}\alpha_0)} P_n(\cos\beta) \right\}. \]

Moreover,

\[ T(\alpha,\beta)_{\alpha=\alpha_0}= \begin{cases} U, & \text{for } 0\le \beta<\beta_0,\\ -U, & \text{for } \beta_0<\beta\le\pi. \end{cases} \]

If the cut passes through the point \(A(\alpha_A,0)\), then the required temperature field is expressed by the equation:

\[ t(\alpha,\beta)= \]

\[ = U \frac{ \displaystyle \sum_{n=1}^{\infty} \bigl[P_{n-1}(\cos\beta_0)-P_{n+1}(\cos\beta_0)\bigr] \left[ \frac{ P_n(i\,\operatorname{sh}\alpha)P_n(\cos\beta)-P_n(i\,\operatorname{sh}\alpha_A) }{ P_n(i\,\operatorname{sh}\alpha_0) } \right] }{ \displaystyle 1+\cos\beta_0- \sum_{n=1}^{\infty} \bigl[P_{n-1}(\cos\beta_0)-P_{n+1}(\cos\beta_0)\bigr] \frac{P_n(i\,\operatorname{sh}\alpha_A)} {P_n(i\,\operatorname{sh}\alpha_0)} }; \]

where \(\operatorname{sh} a_A=\operatorname{sh} x_0 \cos \zeta_0\).

For the case \(x_A=0,\ \zeta_0=\pi/2\),

\[ t(a,\beta)= U \sum_{n=0}^{\infty}(-1)^n \frac{1\cdot 3\cdot 5\cdots (2n-1)(4n+3)} {2\cdot 4\cdot 6\cdots (2n+2)} \frac{P_{2n+1}(i\operatorname{sh} a)} {P_{2n+1}(i\operatorname{sh} a_0)} P_{2n+1}(\cos \beta). \]

References

1. Hobson, E. W. The Theory of Spherical and Ellipsoidal Functions. GIIL, 1952.
2. Lebedev, N. N. Special Functions and Their Applications. Gostekhizdat, 1953.
3. Smirnov, M. M. Problems on the Equations of Mathematical Physics. Gostekhizdat, 1957.
4. Lebedev, N. N.; Skalskaya, I. P.; Uflyand, Ya. S. A Collection of Problems in Mathematical Physics. Gostekhizdat, 1955.