Temperature Field of an Annular Heat-Generating Element under an Arbitrary Time Dependence of the Heat Transfer Coefficient and Ambient Temperature
Abstract Generated abstract
This paper examines transient heat conduction in an annular heat-generating element with time-dependent internal heat generation, heat-transfer coefficients, and ambient temperatures at the inner and outer surfaces. The boundary-value problem is transformed using a finite Hankel integral transform, yielding a formal representation in terms of surface temperatures and a system of functional equations. For the quasi-stationary cooling regime, the authors derive closed analytical relations for the surface temperatures and the temperature field across the annulus, including limiting reductions to solid cylindrical and plane fuel elements. The initial inertial stage is treated approximately with a one-parameter Kármán, Pohlhausen integral method, and comparison with a known semi-infinite-body solution indicates an error not exceeding about 7 percent for the tested case.
Full Text
TEMPERATURE FIELD OF AN ANNULAR HEAT-GENERATING ELEMENT UNDER CONDITIONS OF ARBITRARY TIME DEPENDENCE OF THE HEAT-TRANSFER COEFFICIENT AND AMBIENT TEMPERATURE
E. I. GONCHAROV, V. V. SALOMATOV
(Presented by the scientific seminar of the Department of Boiler Construction and Boiler Installations)
The problem of determining the temperature field of an annular heat-generating element, whose internal heat generation is an arbitrary function of time, under its unsteady cooling by convective heat fluxes under the conditions \(Bi = Bi(Fo)\), \(\Theta_c = \Theta(Fo)\), reduces to integration of the system of differential equations:
\[ \frac{\partial \Theta(x,Fo)}{\partial Fo} = \frac{1}{x}\frac{\partial}{\partial x} \left[ x\frac{\partial \Theta(x,Fo)}{\partial x} \right] + Po(Fo) \tag{1} \]
\[ -\frac{\partial \Theta(1,Fo)}{\partial x} = Bi_1(Fo)\,[\Theta(1,Fo)-\Theta_{c_1}(Fo)] \tag{2} \]
\[ \frac{\partial \Theta(x,Fo)}{\partial x} = Bi_2(Fo)\,[\Theta(x,Fo)-\Theta_{c_2}(Fo)] \tag{3} \]
\[ \Theta(x,0)=\Theta_0=\mathrm{const} \tag{4} \]
\[ \left[ 1\leq x\leq \varkappa;\; 0\leq Fo<\infty;\; \varkappa=\frac{R_2}{R_1} \right] \]
The boundary-value problem (1)–(4) at present has no exact analytical solution because of the structural complexity of the boundary conditions (2), (3). We propose a method whose application makes it possible to obtain analytically exact solutions in the stage of a quasi-stationary cooling regime. The initial stage of the heat-transfer process is calculated approximately by means of the one-parameter Kármán–Pohlhausen method from the theory of the dynamic boundary layer [1, 2].
A formal representation of the temperature field of an annular heat-generating element can be obtained by reducing the boundary-value problem (1)–(4) to a functional equation. Let us apply to the original system of equations the finite Hankel integral transform of type [3]
\[ T(p,Fo)=\int_{1}^{\varkappa}\Theta(x,Fo)M(p,Fo)x\,dx, \tag{5} \]
where
\[ M(p,Fo)=c\left[ I_0(px)\frac{K_1(p)}{I_1(p)}+K_0(px) \right] \tag{6} \]
is the kernel of the integral transform.
Solving the transformed problem and carrying out the transition from the images of the functions to their originals, replacing the modified Bessel functions by ordinary ones, we obtain the formal solution in the form:
\[ \begin{aligned} \Theta(x,Fo) ={}& \Theta_0 + \frac{2\chi^2}{\chi^2-1} \int_0^{Fo} \{ Bi_1(\eta)[\Theta(1,\eta)-\Theta_{c1}(\eta)] \\ &\qquad - \chi Bi_2(\eta)[\Theta(\chi,\eta)-\Theta_{c2}(\eta)] - Po(\eta)\}\,d\eta \\ &{}-\pi \sum_{\mu} \frac{\mu I_1^2(\mu) I_1^2(\chi\mu)} {I_1^2(\mu)-I_1^2(\chi\mu)} \left[ I_0(\mu x)\frac{Y_1(\mu)}{I_1(\mu)} - Y_0(\mu x) \right] \\ &\times \left\{ \int_0^{Fo} \left[ Bi_1(\eta)\frac{\Theta(1,\eta)-\Theta_{c1}(\eta)}{I_1(\mu)} - Bi_2(\eta)\frac{\Theta(\chi,\eta)-\Theta_{c2}(\eta)}{I_1(\mu\chi)} \right.\right.\\ &\qquad\left.\left. - \frac{Po(\eta)}{I_1(\mu)} \right] \exp(\mu^2\eta)\,d\eta \right\} \exp(-\mu^2 Fo). \end{aligned} \tag{7} \]
The eigenvalues of the problem \(\mu\) are found from the equation
\[ I_1(\mu\chi)Y_1(\mu)=Y_1(\mu\chi)I_1(\mu). \tag{8} \]
As follows from (7), the temperature at any point of the heat-generating element can be determined if the temperatures on the boundary surfaces \(\Theta(1,Fo)\) and \(\Theta(\chi,Fo)\) are known.
Putting \(x=1\) and \(x=\chi\) in equation (7) and integrating the last integral by parts, one can obtain the following system of functional equations with respect to \(\Theta(1,Fo)\) and \(\Theta(\chi,Fo)\):
\[ \begin{aligned} \Theta(1,Fo) ={}& \Theta_0 + \frac{2\chi^2}{\chi^2-1} \int_0^{Fo} \{Bi_1(\eta)[\Theta(1,\eta)-\Theta_{c1}(\eta)] \\ &\qquad - \chi Bi_2(\eta)[\Theta(\chi,\eta)-\Theta_{c2}(\eta)] - Po(\eta)\}\,d\eta \\ &{}-M_2(1)\,Bi_1(Fo)[\Theta(1,Fo)-\Theta_{c1}(Fo)] \\ &{}+M_1(1)\,Bi_2(Fo)[\Theta(\chi,Fo)-\Theta_{c2}(Fo)] \\ &{}+\sum_{\mu} Z_1(\mu,1,Fo,\ldots) \end{aligned} \tag{9} \]
\[ \begin{aligned} \Theta(\chi,Fo) ={}& \Theta_0 + \frac{2\chi^2}{\chi^2-1} \int_0^{Fo} \{Bi_1(\eta)[\Theta(1,\eta)-\Theta_{c1}(\eta)] \\ &\qquad - \chi Bi_2(\eta)[\Theta(\chi,\eta)-\Theta_{c2}(\eta)] - Po(\eta)\}\,d\eta \\ &{}-M_2(\chi)\,Bi_1(Fo)[\Theta(1,Fo)-\Theta_{c1}(Fo)] \\ &{}+M_1(\chi)\,Bi_2(Fo)[\Theta(\chi,Fo)-\Theta_{c2}(Fo)] \\ &{}+\sum_{\mu} Z_2(\mu,\chi,Fo,\ldots) \end{aligned} \tag{10} \]
It is known from [4] that, after the lapse of some time \(Fo>Fo^*\), the temperature field enters the stage of an ordered regime. Physically, this means that the temperature gradient at any point of the hollow cylinder remains unchanged, while the character of the temperature distribution over the cross section becomes quasi-steady.
From a mathematical standpoint this indicates the insignificant influence of the last series on the dynamics of the temperature field. Taking into account the indicated circumstances, the system of integral equations (9), (10)
admits a closed-form solution in the stage of the quasi-stationary regime with respect to \(\Theta(1,Fo)\) and \(\Theta(\chi,Fo)\) of the form:
\[ \Theta[((1;\,x),Fo)] = \exp\left(-\int_{0}^{Fo}\frac{P(\eta)}{Q'(\eta)}\,d\eta\right) \left[ \int_{0}^{Fo}G(\eta) \exp\left(\int_{0}^{\eta}\frac{P(\xi)}{Q'(\xi)}\,d\xi\right)d\eta +\Theta_{0} \right]. \tag{11} \]
The values of the quantities entering into (11) are as follows; for \(X=1\)
\[ P(Fo)=\frac{2\chi^{2}}{\chi^{2}-1} \left[ \chi Bi_{2}(Fo)\frac{1-N_{2}Bi_{1}(Fo)}{1-N_{1}Bi_{2}(Fo)} - Bi_{1}(Fo) \right] \tag{12} \]
\[ Q'(Fo)= 1-M_{1}(1)Bi_{2}(Fo)\frac{1-N_{2}Bi_{1}(Fo)}{1-N_{1}Bi_{2}(Fo)} +M_{2}(\chi)Bi_{1}(Fo) \tag{13} \]
\[ \begin{aligned} G(Fo)= -&\left\{ \frac{2\chi^{2}}{\chi^{2}-1}Bi_{1}(Fo)\Theta_{c_1}(Fo)+Po(Fo) \right\} +M_{2}(1)Bi'_{2}(Fo)\times\\ &\times\Theta'_{c_1}(Fo) -\frac{2\chi^{3}}{\chi^{2}-1}Bi_{2}(Fo) \frac{\Theta_{c_2}(Fo)-N_{2}Bi_{1}(Fo)\Theta_{c_1}(Fo)} {1-N_{1}Bi_{2}(Fo)} -\\ &-M_{1}(1)Bi'_{2}(Fo) \left\{ \frac{\Theta_{c_2}(Fo)-N_{2}Bi_{2}(Fo)\Theta_{c_1}(Fo)} {1-N_{1}Bi_{2}(Fo)} \right\} \end{aligned} \tag{14} \]
for \(X=\chi\)
\[ P(Fo)= -\frac{2\chi^{2}}{\chi^{2}-1} \left[ Bi_{1}(Fo)\frac{1-N_{1}Bi_{2}(Fo)}{1-N_{2}Bi_{1}(Fo)} -\chi Bi_{2}(Fo) \right] \tag{15} \]
\[ Q'(Fo)= 1+M_{2}(\chi)Bi_{1}(Fo)\frac{1-N_{1}Bi_{2}(Fo)}{1-N_{2}Bi_{1}(Fo)} -M_{1}(\chi)Bi_{2}(Fo) \tag{16} \]
\[ \begin{aligned} G(Fo)=& -\frac{2\chi^{3}}{\chi^{2}-1}Bi_{2}(Fo)\Theta_{c_2}(Fo) +\frac{2\chi^{2}}{\chi^{2}-1} \left\{ Bi_{1}(Fo)\times \frac{\Theta_{c_1}(Fo)-N_{1}Bi_{2}(Fo)\Theta_{c_2}(Fo)} {1-N_{2}Bi_{1}(Fo)} +Po(Fo) \right\}\\ &-M_{1}(\chi)Bi'_{2}(Fo)\times\Theta'_{c_2}(Fo) +M_{2}(\chi)Bi'_{1}(Fo) \left\{ \frac{\Theta_{c_1}(Fo)-N_{1}Bi_{2}(Fo)\Theta_{c_2}(Fo)} {1-N_{2}Bi_{1}(Fo)} \right\}' \end{aligned} \tag{17} \]
The values of the coefficients \(M_{1}\), \(M_{2}\), \(N_{1}\), \(N_{2}\) are given in Table 1.
Table 1
Values of the coefficients in equations (12)–(17)
| Coefficient | Value |
|---|---|
| \(M_{1}(x)\) | \(-\dfrac{\chi^{3}}{\chi^{2}-1}\left[\dfrac{1}{4}(\chi^{2}-2x^{2})+\left(\ln x+\dfrac{\chi^{4}}{\chi^{2}-1}\ln\dfrac{1}{\chi}+\dfrac{3}{4}\right)\right]\) |
| \(M_{2}(x)\) | \(-\dfrac{\chi^{2}}{\chi^{2}-1}\left[\dfrac{1}{4}(1-2x^{2})+x^{2}\left(\ln\dfrac{x}{\chi}+\dfrac{\chi^{2}}{\chi^{2}-1}\ln\dfrac{1}{\chi}+\dfrac{3}{4}\right)\right]\) |
| \(N_{1}\) | \(M_{1}(\chi)-M_{1}(1)\) |
| \(N_{2}\) | \(M_{2}(\chi)-M_{2}(1)\) |
From the known surface temperatures (relation (11)), it is now possible to calculate the temperature field in any cross section of the annular fuel element.
\[ \begin{aligned} \Theta(x,Fo)=\Theta_0+\frac{2\chi^2}{\chi^2-1}\Bigg\{& \int_0^{Fo}\Big[Bi_1(\eta)\,[\Theta(1,\eta)-\Theta_{c1}(\eta)] \\ &-\chi Bi_2(\eta)\,[\Theta(\chi,\eta)-\Theta_{c2}(\eta)]-Po(\eta)\Big]\,d\eta \\ &-\left[\frac14\left(1-\frac{2}{\chi^2}x^2\right) +\frac{1}{\chi^2}\left(\ln x+\frac{\chi^2}{\chi^2-1}\ln\frac{1}{\chi}+\frac34\right)\right] \\ &\times \Big\{Bi_1(Fo)[\Theta(1,Fo)-\Theta_{c1}(Fo)] \\ &-\chi Bi_2(Fo)[\Theta(\chi,Fo)-\Theta_{c2}(Fo)]-Po(Fo)\Big\}. \end{aligned} \tag{18} \]
As a particular case of the general solution (18), setting \(R_1 \to 0\) and adopting the corresponding boundary condition
\[ \frac{\partial \Theta(0,Fo)}{\partial x}=0, \]
one can derive the calculation relations obtained by us earlier [5]. By setting \(R_1\) and \(R_2 \to \infty\), the problem reduces to nonsymmetric cooling of a plane fuel element.
To obtain calculation relations at the initial stage, we shall use the approximate integral method of Karman—Pohlhausen from boundary-layer theory.
By analogy with the thickness of the dynamic boundary layer, let us introduce the thickness of the thermal layer \(\delta_1(Fo)=\xi_1(Fo)-1\) for the inner part and \(\delta_2(Fo)=\chi-\xi_2(Fo)\) for the outer part of the nonsymmetrically cooled fuel element.
In accordance with the adopted solution scheme, the boundary conditions (2), (3) for the differential equation (1), which now, for the inner and outer parts of the fuel element, as for semi-infinite regions, is replaced by integral relations,
\[ \frac{\partial}{\partial Fo} \left[ \int_{1}^{\xi_1}\Theta_1(x,Fo)\,dx-Q(Fo)\xi_1 \right] +\frac{\partial\Theta(1,Fo)}{\partial x} -\frac{\partial\Theta(\xi_1,Fo)}{\partial x}=0, \tag{19} \]
\[ \frac{\partial}{\partial Fo} \left[ \int_{\xi_2}^{\chi}\Theta_2(x,Fo)\,dx-Q(Fo)\xi_2 \right] +\frac{\partial\Theta(\chi,Fo)}{\partial x} -\frac{\partial\Theta(\xi_2,Fo)}{\partial x}=0, \tag{20} \]
where
\[ Q(Fo)=\int_0^{Fo}Po(\eta)\,d\eta, \]
are modified somewhat, namely,
\[ \Theta_1(\xi_1,Fo)=Q(Fo),\qquad \Theta_2(\xi_2,Fo)=Q(Fo) \tag{21} \]
\[ \frac{\partial\Theta_1(\xi_1,Fo)}{\partial x}=0,\qquad \frac{\partial\Theta_2(\xi_2,Fo)}{\partial x}=0. \tag{22} \]
We shall approximate the temperature field in the zone of thermal disturbance by a third-degree polynomial in \(X\)
\[ \Theta_1(x,Fo)-\Theta_c =C_1(Fo)+C_2(Fo)(\xi_1-x)+C_3(Fo)(\xi_1-x)^2+ \]
\[ +C_4(Fo)(\xi_1-x)^3 \tag{23} \]
\[
\Theta_2(x,Fo)-\Theta_0
= D_1(Fo)+D_2(Fo)(x-\xi_2)+
\]
\[
{}+D_3(Fo)(x-\xi_2)^2+D_4(Fo)(x-\xi_2)^3,
\tag{24}
\]
where \(C_i(Fo)\), \(D_i(Fo)\) are, for the time being, unknown functions of dimensionless time.
To determine the four coefficients of the cubic parabola, three conditions, (4), (21), (22), are already available. The missing condition may be written, following [1], in the form
\[ \frac{\partial^2\Theta_1(\xi_1,Fo)}{\partial x^2}=0,\qquad \frac{\partial^2\Theta_2(\xi_2,Fo)}{\partial x^2}=0. \tag{25} \]
Determining the unknown coefficients and integrating the heat-balance equations (19), (20), we obtain differential equations for determining the thicknesses of the thermal layers
\[
\frac{\partial}{\partial Fo}
\left\{
\frac{Bi_1(Fo)\,[\Theta_{c_1}(Fo)-Q(Fo)]\,\delta_1^2(Fo)}
{3+Bi_1(Fo)\delta_1(Fo)}
\right\}
=
\]
\[
=12\,\frac{Bi_1(Fo)\,[\Theta_{c_1}(Fo)-Q(Fo)]}
{3+Bi_1(Fo)\delta_1(Fo)}
\tag{26}
\]
\[
\frac{\partial}{\partial Fo}
\left\{
\frac{Bi_2(Fo)\,[\Theta_{c_2}(Fo)-Q(Fo)]\,\delta_2^2(Fo)}
{3+Bi_2(Fo)\delta_2(Fo)}
\right\}
=
\]
\[
=12\,\frac{Bi_2(Fo)\,[\Theta_{c_2}(Fo)-Q(Fo)]}
{3+Bi_2(Fo)\delta_2(Fo)}
\tag{27}
\]
under the condition \(\delta_i(0)=0\).
It is difficult to find the general solution of the obtained equations (26), (27), but for two cases the solutions reduce to quadratures:
1) \(Bi(Fo)\delta_i(Fo)\ll 3\)
\[ \delta_i= \left\{ \frac{12}{Bi_i|\Theta_{c_i}-Q|} \int_0^{Fo} Bi_i|\Theta_{c_i}-Q|\,d\eta \right\}^{1/2} \tag{28} \]
2) \(Bi(Fo)\delta_i(Fo)\gg 3\)
\[ \delta_i= \left\{ \frac{24}{|\Theta_{c_i}-Q|^2} \int_0^{Fo} [\Theta_{c_i}-Q]^2\,d\eta \right\}^{1/2}. \tag{29} \]
Calculation by the approximate relations obtained may be carried out until
\[ \delta_1(Fo)+\delta_2(Fo)\le \chi-1. \tag{30} \]
To estimate the accuracy of reproducing the temperature field, the exact solution for the case of cooling of a semi-bounded body in a medium with \(a=\mathrm{const}\), \(\Theta_c=\mathrm{const}\), \(Po=0\) [6] was compared with the approximate solution by the proposed method. As this comparison showed, the error in determining the temperature field does not exceed 7% and can be reduced by increasing the degree of the approximating polynomial.
For the transition zone between the initial cooling period and the quasi-steady stage of the process, the solution may be obtained approximately by expanding the solution at \(Fo=Fo^*\) or \(Fo=Fo^{**}\) in a Taylor series.
Conclusions
- Solutions (11), (18) are analytically exact for the quasi-steady terms of the problem under study.
- The inertial period is calculated approximately from relations (23), (24), obtained by the Kármán—Pohlhausen method from boundary-layer theory.
References
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- T. Goodman. In: Problems of Heat Transfer, Atomizdat, 1967.
- E. P. Trofimov. IFZh, 1961, vol. III, No. 10.
- A. V. Lykov. In: Hardening of Metals, Mashgiz, 1958.
- V. V. Salomatov, E. I. Goncharov. Proceedings of the USSR Academy of Sciences, “Power Engineering and Transport,” 1967, No. 4.
- A. V. Lykov. Theory of Heat Conduction. Gostekhizdat, 1952.