Nonstationary Temperature Fields in Shells under Their High-Temperature Heating Regime

V. V. Salomatov

(Presented by the scientific seminar of the Heat-Power Engineering Faculty)

The article investigates the process of nonstationary heat transfer in shells, when the heat fluxes forming the temperature field are connected with the corresponding surface temperatures by arbitrary nonlinear dependences. Analytically exact solutions are constructed for the quasi-stationary terms of the problem. The transient process is investigated by means of a one-parameter method from boundary-layer theory.

High-temperature heating of shells of various geometric forms, for most practical heat-transfer conditions, can be formulated mathematically as the boundary-value problem

\[ \frac{1}{\xi^\nu}\frac{\partial}{\partial \xi} \left(\xi^\nu \frac{\partial \Theta}{\partial \xi}\right) = \frac{\partial \Theta}{\partial Fo}, \tag{1} \]

\[ \Theta(\xi,0)=\Theta_0, \tag{2} \]

\[ \left. \frac{\partial \Theta}{\partial \xi} \right|_{\xi=\xi_1} = f_1[\Theta(\xi_1,Fo)], \tag{3} \]

\[ \left. \frac{\partial \Theta}{\partial \xi} \right|_{\xi=\xi_2} = f_2[\Theta(\xi_2,Fo)], \tag{4} \]

where \(\nu\) is a form constant, numerically equal to \(0, 1, 2\), respectively, for plane, cylindrical, and spherical shells; \(f_1, f_2\) are arbitrary integrable functions of the surface temperatures. Owing to the structural complexity of the boundary conditions (3)—(4), it is not possible to obtain a solution of the boundary-value problem (1)—(4) in explicit form. In connection with this, the present work proposes a method of approximate solution of the indicated problem, which makes it possible to obtain the most rational, from a practical standpoint, computational relations for large and small values of the dimensionless time.

1. The temperature field of a plane shell \((\nu=0,\ \xi_1=1,\ \xi_2=0)\). Applying the Laplace integral transform to (1)—(4), we obtain the solution of the transformed problem

\[ W(\xi,s)=\frac{\Theta_0}{s} + U_1(1,s)\frac{\operatorname{ch}\sqrt{s}\,\xi}{\sqrt{s}\operatorname{sh}\sqrt{s}} - U_2(0,s)\frac{\operatorname{ch}\sqrt{s}(\xi-1)}{\sqrt{s}\operatorname{sh}\sqrt{s}}. \tag{5} \]

Here

\[ W(\xi,s)=\int_0^\infty \Theta(\xi,Fo)\exp(-sFo)\,dFo, \]

\[ U_1(1,s)=\int_0^\infty f_1[\Theta(1,\mathrm{Fo})]\exp(-s\mathrm{Fo})\,d\mathrm{Fo}, \]

\[ U_2(0,s)=\int_0^\infty f_2[\Theta(0,\mathrm{Fo})]\exp(-s\mathrm{Fo})\,d\mathrm{Fo}. \]

Carrying out the inverse transformation, we present the formal solution in the form

\[ \begin{aligned} \Theta(\xi,\mathrm{Fo})=\Theta_0 &+\int_0^{\mathrm{Fo}}\{f_1[\Theta(1,\eta)]+f_2[\Theta(0,\eta)]\}\,d\eta \\ &-\frac{1}{6}f_1[\Theta(1,\mathrm{Fo})](1-3\xi^2) -\frac{1}{6}f_2[\Theta(0,\mathrm{Fo})][1-3(1-\xi)^2] \\ &+\sum_{n=1}^{\infty}\frac{2(-1)^n}{\mu_n^2}\Bigg\{ \cos\mu_n\xi\int_0^{\mathrm{Fo}} f_1'[\Theta(1,\eta)]\exp[-\mu_n^2(\mathrm{Fo}-\eta)]\,d\eta \\ &\qquad\qquad +\cos\mu_n(1-\xi)\int_0^{\mathrm{Fo}} f_2'[\Theta(1,\eta)]\exp[-\mu_n^2(\mathrm{Fo}-\eta)]\,d\eta \Bigg\}. \end{aligned} \tag{6} \]

Putting successively \(\xi=1\) and \(\xi=0\) in (6), one can obtain a system of two nonlinear functional equations of the Volterra type of the second kind for determining the surface temperatures. Proving absolute and uniform convergence of the infinite series, using, for example, the integral estimates in [1], we arrive at the assertion of convergence of the Cauchy—Picard successive-approximation method in solving the indicated integral equations. As zero approximations it is expedient to choose the free terms of the functional equations obtained. However, in order to obtain computational relations most convenient for engineering practice, whose structure would not contain sums of infinite series, the solutions are carried out separately for the quasi-stationary and transient periods.

Owing to the inequalities \(\mu_1<\mu_2<\mu_3,\ldots\) and the presence of exponential-type factors, the sum of the series (6), beginning with some value of \(\mathrm{Fo}\), becomes negligibly small in comparison with the remaining terms. From this moment the stage of a quasi-established regime begins. The surface temperatures for this stage are found from the relations

\[ \mathrm{Fo}=\int_{\Theta_*(1,0)}^{\Theta(1,\mathrm{Fo})} \frac{1-\frac{1}{3}f_1'(\Theta)+\frac{1}{6}[f_2^{+}(\Theta)]'} {f_1(\Theta)+f_2^{+}(\Theta)}\,d\Theta, \tag{7} \]

where \(\Theta_*(1,0)\) is determined as the real root of the equation

\[ \Theta_*(1,0)=\Theta_0+\frac{1}{3}f_1[\Theta_*(1,0)]-\frac{1}{6}f_2^{+}[\Theta_*(1,0)] \]

and

\[ f_1[\Theta(1,\mathrm{Fo})]-f_2[\Theta(0,\mathrm{Fo})]=2[\Theta(1,\mathrm{Fo})-\Theta(0,\mathrm{Fo})]. \tag{8} \]

Knowing the laws of variation of the temperatures on the boundary surfaces, one can also calculate the entire temperature field in the quasi-established period

\[ \Theta(\xi,\mathrm{Fo})=\Theta(1,\mathrm{Fo}) -\frac{1}{6}f_1[\Theta(1,\mathrm{Fo})][2+(1-3\xi^2)]- \]

\[ -\frac{1}{6}f_2[\Theta(0,\mathrm{Fo})][2-3(1-\xi)^2]. \tag{9} \]

The initial stage of the heat-transfer process can be found by the method of boundary-layer theory. In accordance with the general theory

of the integral method, the concept of the thickness of the thermal layer is introduced [2]. In our case, \(\delta_1\) is for the inner part and \(\delta_2\) for the outer part of the shells. The temperature profile in the zone of thermal disturbance will be approximated by a third-degree polynomial in \(\xi\), for the inner part,

\[ \Theta_1(\xi,\mathrm{Fo}) = C'_1 + C'_2(\delta_1-\xi) + C'_3(\delta_1-\xi)^2 + C'_4(\delta_1-\xi)^3, \tag{10} \]

for the outer part,

\[ \Theta_2(\xi,\mathrm{Fo}) = C''_1 + C''_2(\delta_2-(1-\xi)) + C''_3(\delta_2-(1-\xi))^2 + C''_4(\delta_2-(1-\xi))^3. \tag{11} \]

To determine the eight unknown coefficients in (10)—(11), we shall use the following boundary conditions

\[ \Theta(\xi,0)\big|_{\xi=\delta_1,\delta_2}=\Theta_0;\qquad \frac{\partial \Theta(\xi,\mathrm{Fo})}{\partial \xi}\bigg|_{\xi=\delta_1,\delta_2}=0;\qquad \frac{\partial^2 \Theta(\xi,\mathrm{Fo})}{\partial \xi^2}\bigg|_{\xi=\delta_1,\delta_2}=0; \]

as well as conditions (3), (4).

Taking into account the found values of the coefficients of the approximating polynomials, relations (10)—(11) can be written in the form

\[ \Theta_1(\xi,\mathrm{Fo}) = \Theta_0 + \frac{f_1[\Theta(1,\mathrm{Fo})]}{3\delta_1^2}\,|\delta_1-\xi|^3, \tag{12} \]

\[ \Theta_2(\xi,\mathrm{Fo}) = \Theta_0 + \frac{f_2[\Theta(0,\mathrm{Fo})]}{3\delta_2^2}\,|\delta_2-(1-\xi)|^3. \tag{13} \]

The unknown thickness of the thermal layer \(\delta_1\) or \(\delta_2\) is related to the corresponding surface temperature by the relation

\[ \delta_i=\frac{3(\Theta_{\mathrm{surf}\,i}-\Theta_0)} {f_i[\Theta_{\mathrm{surf}\,i}]}. \tag{14} \]

Integrating the differential equation of the process (1) over the thickness of the corresponding thermal layer, one can determine, from the obtained integral of the heat balance, the unknown surface temperature. Carrying out the indicated operations and solving the resulting differential equations, we find that the unknown temperatures at the boundary surfaces are calculated from the following expressions

\[ \frac{4}{3}\,\mathrm{Fo}= \int_{\Theta_0}^{\Theta(1,\mathrm{Fo})} \frac{2(\Theta-\Theta_0)f_1(\Theta)-f'_1(\Theta)(\Theta-\Theta_0)^2} {f_1^3(\Theta)}\,d\Theta, \tag{15} \]

\[ \frac{4}{3}\,\mathrm{Fo}= \int_{\Theta_0}^{\Theta(0,\mathrm{Fo})} \frac{2(\Theta-\Theta_0)f_2(\Theta)-f'_2(\Theta)(\Theta-\Theta_0)^2} {f_2^3(\Theta)}\,d\Theta. \tag{16} \]

The integrals in relations (15), (16), for most conditions of external heat exchange, are, as a rule, representable in elementary functions.

The calculation by relations (12), (13) can be carried out until the sum of the thicknesses of the thermal layers exceeds the shell thickness, which is expressed analytically in the form

\[ \frac{3|\Theta(1,\mathrm{Fo})-\Theta_0|}{f_1[\Theta(1,\mathrm{Fo})]} + \frac{3|\Theta(0,\mathrm{Fo})-\Theta_0|}{f_2[\Theta(0,\mathrm{Fo})]} \leqslant 1. \]

2. Temperature field of a cylindrical shell

\[ \left( \nu=1,\qquad \xi_1=1,\qquad \xi_2=p,\qquad p=\frac{R_1}{R_2} \right). \]

Transforming equations (1)—(4) by means of the Laplace transform into the image domain and solving the transformed problem, we obtain

\[ W(\xi,s)=\frac{\Theta_0}{s}+U_1(1,s)\frac{I_0(\sqrt{s}\xi)}{\sqrt{s}I_1(\sqrt{s})} -U_2(p,s)\frac{I_0[\sqrt{s}(\xi-p)]}{\sqrt{s}I_1(\sqrt{s})}. \tag{17} \]

Passing in (17) from images to the originals of the functions, we write the final solution in the form

\[ \begin{aligned} \Theta(\xi,\mathrm{Fo})=\Theta_0 &+\frac{2}{1-p^2}\left\{\int_0^{\mathrm{Fo}}\left[f_2(\Theta(p,\vartheta))+f_1(\Theta(1,\vartheta))\right]\,d\vartheta\right\} \\ &-\frac{1}{1-p^2}\left[\frac14(1-2\xi^2)+p^2\left(\ln\xi+\frac{1}{1-p^2}\ln p+\frac34\right)\right]f_2[\Theta(p,\mathrm{Fo})] \\ &-\frac{p^2}{1-p^2}\left\{\frac14\left[1-2(p-\xi)^2\right]+p^2\left[\ln\frac{\xi}{p}+\frac{1}{1-p^2}\ln p+\frac34\right]\right\}f_1[\Theta(1,\mathrm{Fo})] \\ &+\sum_{n=1}^{\infty}\frac{\pi}{\mu_n^2}\cdot \frac{J_1(\mu_np)J_1(\mu_n)}{J_1^2(\mu_np)-J_1^2(\mu_n)} \left[J_0(\mu_n\xi)Y_1(\mu_np)-Y_0(\mu_n\xi)J_1(\mu_np)\right]\times \\ &\times \int_0^{\mathrm{Fo}} \left[f_2'[\Theta(p,\vartheta)]+f_1'[\Theta(1,\vartheta)]\right] \exp\left[-\mu_n^2(\mathrm{Fo}-\vartheta)\right]\,d\vartheta, \end{aligned} \tag{18} \]

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

\[ J_1(\mu_np)Y_1(\mu_n)=Y_1(\mu_np)J_1(\mu_n). \]

Putting \(\xi=1\) and \(\xi=p\) in (18), we see that the unknown temperatures on the surfaces of the cylindrical shell satisfy a system of nonlinear functional equations. A closed solution of the obtained system of integral equations is possible only at the stage of the ordered regime. Similarly, the surface temperature functions are found from the following relations:

\[ \frac{1-p^2}{2}\mathrm{Fo}= \int_{\Theta_*(1,0)}^{\Theta(1,\mathrm{Fo})} \frac{1-N(p)[f_2^+(\Theta)]'+M(p)f_1'(\Theta)} {f_1(\Theta)+f_2^+(\Theta)}\,d\Theta, \]

\[ N(p)= \frac{\dfrac14-p^2\left(\dfrac{\ln p}{1-p^2}+\dfrac34\right)} {1-p^2}, \]

\[ M(p)= \frac{p^2\left[\dfrac14(1-2(p-1)^2)+\ln\frac1p+\dfrac{\ln p}{1-p^2}+\dfrac34\right]} {1-p^2}, \tag{19} \]

where \(\Theta_*(1,0)\) is found as the root of the algebraic equation

\[ \Theta_*(1,0)=\Theta_0+N(p)f_2^+[\Theta_*(1,0)]-M(p)f_1[\Theta_*(1,0)] \]

and

\[ \Theta(1,\mathrm{Fo})-\Theta(p,\mathrm{Fo})= \frac{\dfrac12(1-p^2)+p^2\ln p}{1-p^2} \left\{f_2[\Theta(p,\mathrm{Fo})]+p^2f_1[\Theta(1,\mathrm{Fo})]\right\}. \tag{20} \]

From the surface temperatures found, on the basis of (17) one can determine the temperature field in any section of the shell at the stage of the quasi-stationary regime.

\[ \Theta(\xi,\mathrm{Fo})=\Theta(1,\mathrm{Fo})-\frac{f_2[\Theta(p,\mathrm{Fo})]}{1-p^2} \left[\frac14(1-2\xi^2)+p^2\ln \xi+\frac14\right]- \]
\[ -\frac{p^2 f_1[\Theta(1,\mathrm{Fo})]}{1-p^2} \left[p^2\ln \xi-\frac12(p-\xi)^2+\frac12(1-p)^2\right]. \tag{21} \]

The initial stage can also be calculated approximately by means of the Karman—Pohlhausen integral method.

We approximate the unknown temperature profiles in the region of thermal disturbances by the cubic dependences (12)—(13). The unknown temperatures at the outer boundaries of the cylindrical shell are found analogously from the heat-balance integral and are calculated from the relations

\[ \frac{8}{3}\mathrm{Fo}= \int_{\Theta_0}^{\Theta(1,\mathrm{Fo})} \frac{2(\Theta-\Theta_0)f_1(\Theta)-f'_1(\Theta)(\Theta-\Theta_0)^2} {f_1^3(\Theta)}\,d\Theta \tag{22} \]

\[ \frac{8}{3}\mathrm{Fo}= \int_{\Theta_0}^{\Theta(p,\mathrm{Fo})} \frac{2(\Theta-\Theta_0)f_2(\Theta)-f'_2(\Theta)(\Theta-\Theta_0)^2} {f_2^3(\Theta)}\,d\Theta. \tag{23} \]

The problem for a spherical shell is solved in an entirely analogous manner.

REFERENCES

1. V. V. Salomatov, E. I. Goncharov. Izvestiya AN SSSR, “Energetika i transport,” No. 4, 1967.
2. G. Schlichting. Boundary-Layer Theory. IL Publishing House, 1956.