On asymmetric heating of a flat wall
Abstract Generated abstract
This paper addresses transient heating or cooling of an infinite flat wall subjected to different convective conditions on its two faces, a problem whose exact analytical solutions are described as cumbersome for practical use. The authors propose approximate temperature field expressions constructed from known symmetric heating solutions, supplemented by an auxiliary term chosen to satisfy the initial condition and the steady-state limiting temperature gradient. They discuss the expected accuracy of the formulas for equal and unequal Biot numbers, compare one calculated temperature profile with a numerical integration result, and indicate a modified expression for small Biot numbers. The approach is presented as a simpler calculation method that can be extended when symmetric solutions are available for the relevant boundary conditions.
Full Text
On Asymmetric Heating of a Flat Wall
Yu. V. Vidin, G. N. Boikov
(Presented by Professor, Doctor G. I. Fuks)
If it is assumed that the thermophysical characteristics of a substance do not change in the process of heating (cooling) of a solid body, and if the process itself takes place under the action of convective heat exchange obeying Newton’s law, then the system of differential equations describing the process may be represented as follows:
\[ \frac{\partial t(x,\tau)}{\partial \tau} = a \cdot \frac{\partial^2 t(x,\tau)}{\partial x^2}, \tag{1} \]
\[ \lambda \cdot \frac{\partial t(+R,\tau)}{\partial x} = \alpha_1 \cdot \left[t_{c1} - t(+R,\tau)\right], \tag{2} \]
\[ -\lambda \cdot \frac{\partial t(-R,\tau)}{\partial x} = \alpha_2 \cdot \left[t_{c2} - t(-R,\tau)\right], \tag{3} \]
\[ t(x,0)=t_0, \tag{4} \]
where
\(t_{c1}; \alpha_1\) — the temperature and heat-transfer coefficient of the first medium;
\(t_{c2}; \alpha_2\) — the temperature and heat-transfer coefficient of the second medium;
\(2R\) — the wall thickness (the origin of coordinates is at the center of the plate).
The solution of such a system presents certain difficulties, since in this case the solution of the differential equation has to be matched with the boundary conditions on two sides. In the literature there are solutions of a similar asymmetric problem; however, they have a very complex and cumbersome final result, whose practical application is associated with great difficulties [1, 2, 5].
In our opinion, in a number of cases it is expedient to approach the investigation of this question from somewhat different positions, namely, proceeding from physical and quantitative conceptions of the process of asymmetric heating.
The problem of heating (cooling) under asymmetric boundary conditions can, with known accuracy, be solved on the basis of available solutions of symmetric problems, taking into account the physical picture of the process and certain quantitative relations. Proceeding from what has been said, it is proposed to carry out the calculation of the temperature
field described by the system of equations (1), (2), (3), and (4), on the basis of the expression
\[ t(x,\tau)=0.5\,[t_{\mathrm{sym.I}}(x,\tau)+t_{\mathrm{sym.II}}(x,\tau)]+ \]
\[ +\kappa\,(x+l)\left[1-\sum_{1}^{\infty} A_n^0\cdot \cos \psi_n^0\cdot e^{-(\psi_n^0)^2\cdot Fo}\right], \tag{5} \]
where \(t_{\mathrm{sym.I}}(x,\tau)\) is the solution of the symmetric problem at medium temperature \(t_{c1}\) and heat-transfer coefficient \(\alpha_1\).
\(t_{\mathrm{sym.II}}(x,\tau)\) is analogous for the second medium \((t_{c2}; \alpha_2)\);
\[ \kappa \]
is a constant coefficient equal to:
\[ \kappa=\frac{t_{c1}-t_{c2}}{2R}\cdot \frac{2Bi_1\cdot Bi_2}{2Bi_1\cdot Bi_2+Bi_1+Bi_2}. \]
The values of the constants \(A_n^0\) and the roots \(\psi_n^0\) are determined from the arithmetic-mean value of the Biot criterion
\[ \left(Bi_0=\frac{Bi_1+Bi_2}{2}\right) \]
[3]. The quantity \(l\) is determined from the relation:
\[ l=R\cdot \frac{Bi_1-Bi_2}{2Bi_1\cdot Bi_2}. \]
The first term \(0.5\,[t_{\mathrm{sym.I}}(x,\tau)+t_{\mathrm{sym.II}}(x,\tau)]\) (the principal term of the expression) gives the temperature field in the plate under averaged boundary conditions. Therefore it is natural to expect that the results of calculating this half-sum will prove sufficiently accurate for the central layers of an infinite flat wall throughout the entire heating period. It is quite clear that the temperatures at the surfaces will in reality differ from those obtained from the principal term of the expression. These deviations are taken into account by the second term (the auxiliary term of the expression).
The form (structure) of the auxiliary term was adopted on the basis of the following considerations:
- The initial conditions (4) must be exactly satisfied.
- The final conditions of the process must be strictly satisfied (the temperature distribution in the system at \(\tau=\infty\)).
On the basis of these conditions, the value of the constant \(\kappa\) was determined as the temperature gradient under steady-state conditions, to which the system ultimately comes.
The proposed relation (5) has the following features.
For equal values of the heat-transfer coefficients \((Bi_1=Bi_2)\)
1) it satisfies the differential equation (1) with an accuracy of up to 10% at \(Bi_1=Bi_2=\infty\) and
\[ p=\frac{t_{c2}-t_0}{t_{c1}-t_0}=0.7; \]
as the values of the Biot criteria decrease and the value of the complex \(p\) increases, the discrepancy between relation (5) and differential equation (1) decreases and tends to zero;
2) it satisfies the boundary conditions (2) and (3) with an accuracy of up to 10% at \(Bi_1=Bi_2=0.01\) and \(p=0.7\); as the values of the Biot criteria increase and the value of the complex \(p\) increases, the discrepancy between relation (5) and the boundary conditions decreases and tends to zero.
For different values of the heat-transfer coefficients \((Bi_1\ne Bi_2)\), relation (5) satisfies the boundary conditions (2) and (3) with an accuracy of up to 10% when the values of \(Bi_1\) and \(Bi_2\) differ by a factor of 2–3 and \(p=0.7\). As the difference between the Biot criteria decreases, the agreement improves.
and the discrepancy tends to zero. Under the same conditions, expression (5) agrees better with the differential equation (1).
Figure 1 shows the pattern of the temperature field obtained from calculations by expression (5). Also plotted here, by points, is the curve of the temperature field obtained from the solution of the system of equations (1), (2), (3), and (4) by the method of numerical integration [4] under the same conditions.
For small values of the Biot criteria (0.5 and less), the more accurate solution will be the following:
\[ t(x,\tau)=t^{0}_{\mathrm{sym}}(x,\tau)+\kappa(x+l)\cdot \left[1-\sum_{1}^{\infty} A^{0}_{n}\cos\mu^{0}_{n}\cdot e^{-(\mu^{0}_{n})^{2}Fo}\right]; \tag{6} \]
where \(t^{0}_{\mathrm{sym}}(x,\tau)\) is the solution of the symmetric problem at the medium temperature
\[ t_{c0}=\frac{t_{c1}+t_{c2}}{2} \]
and heat-transfer coefficient
\[ \alpha_0=\frac{\alpha_1+\alpha_2}{2}. \]
The second term is the same as in relation (5).
Fig. 1. Temperature field of an unbounded plate
(\(Bi_1=3;\ Bi_2=1;\ t_{c1}=1020^\circ\mathrm{C};\ t_{c2}=720^\circ\mathrm{C};\ t_0=20^\circ\mathrm{C}.\))
Thus, the proposed expressions (5) and (6) are composed from symmetric solutions available in the literature. This, first, facilitates carrying out the calculation and, second, creates the prerequisite for the technical possibility of establishing a comparatively simple dependence under any boundary conditions for which symmetric solutions exist.
The verification of formulas (5) and (6) was carried out for conditions of the steady-state regime.
REFERENCES
- G. V. Plyatsko. IFZh, No. 10, 1959.
- G. I. Pavlovskii. IFZh, No. 4, 1962.
- A. V. Lykov. Theory of Heat Conduction, Gostekhizdat, 1952.
- A. P. Vanichev. Proceedings of NII-1, No. 25, 1947.
- B. F. Glikman, ZhTF, vol. 27, issue 12, 1957.