On the Cooling of a Horizontal Infinitely Long Cylinder by Unsteady Natural Convection

L. I. Kudryashev, V. V. Banyukha

An approximate method is proposed for solving the system of equations describing the process of cooling a horizontal infinitely long cylinder under conditions of unsteady natural convection. The thermophysical characteristics of the body and of the medium are assumed to be independent of temperature.

Cases in which the temperature of the medium is constant, while the body introduced into it has, at the initial moment of time, a temperature noticeably different from it, are of great practical interest. In this case the temperature of the body will asymptotically approach the temperature of the medium. This case of the so-called “simple” cooling of a cylinder, which, along with the sphere and the plate, is an elementary form of the surface of heat-exchange apparatuses, was used by us to determine the laws governing the behavior of the heat-transfer coefficient under conditions of unsteady natural convection.

The problem has the following mathematical formulation:

\[ \frac{\partial w_x}{\partial x}+\frac{\partial w_y}{\partial y}=0, \tag{a} \]

\[ \frac{\partial w_x}{\partial \tau} +w_x\frac{\partial w_x}{\partial x} +w_y\frac{\partial w_y}{\partial y} = \nu\frac{\partial^2 w_x}{\partial y^2} + g\,\frac{t-t_\infty}{t_\infty}\,\sin\beta, \tag{b} \]

\[ \frac{\partial t}{\partial \tau} +w_x\frac{\partial t}{\partial x} +w_y\frac{\partial t}{\partial y} = \bar a\,\frac{\partial^2 t}{\partial y^2}, \tag{c} \]

\[ -\,c_{pm}\rho_m\,\frac{D}{4}\,\frac{\partial t_w}{\partial \tau} = \alpha\,(t_w-t_\infty). \tag{d} \]

\[ \tag{1} \]

When using the region of influence “\(\Delta\),” it is assumed that \(w_y=0\). If we introduce the dimensionless temperature defined by the expression

\[ \Theta=\frac{t-t_\infty}{t_{w,\max}-t_\infty}, \tag{2} \]

then system (1) will have the form:

\[ \frac{\partial w_x}{\partial x}=0, \tag{a} \]

\[ \frac{\partial w_x}{\partial \tau} = \nu \frac{\partial^2 w_x}{\partial y^2} + g\,\frac{t_{w,\max}-t_\infty}{t_\infty}\,\Theta \sin \beta, \tag{b} \]

\[ \frac{\partial \Theta}{\partial \tau} + w_x \frac{\partial \Theta}{\partial x} = a \frac{\partial^2 \Theta}{\partial y^2}, \tag{c} \]

\[ - c_{pM}\rho_M \frac{D}{4}\,\frac{d\Theta_w}{d\tau} = \alpha \Theta_w, \tag{d} \]

\[ \tag{3} \]

where

\[ \Theta_w=\frac{t_w-t_\infty}{t_{w,\max}-t_\infty}. \tag{4} \]

Equation (3b) belongs to the class of nonhomogeneous linear equations. It is solved under the following initial and boundary conditions:

\[ \begin{array}{lll} \text{at } \tau=0 & w_x=0, & \tag{a}\\[2mm] y=0 & w_x=0, & \tag{b}\\[2mm] y=\Delta & w_x=0. & \tag{c} \end{array} \tag{5} \]

The solution of equation (3b) under the boundary conditions (5) is known [1] and has the form

\[ w_x(y,\tau) = \frac{2}{\Delta} \sum_{n=1}^{\infty} \left\{ \int_0^\tau e^{-\left(\frac{n\pi}{\Delta}\right)^2(\tau-\varepsilon)} \int_0^\Delta g\,\frac{t_{w,\max}-t_\infty}{t_\infty}\, \Theta(\varepsilon)\sin\beta\,d\varepsilon \right\} \sin\left(\frac{n\pi y}{\Delta}\right). \tag{6} \]

However, its further use is difficult. Below a scheme is proposed for a simplified solution of the problem. We shall seek the solution of equation (3b) in the form of a function satisfying both the initial and boundary conditions (5):

\[ w_x = g\,\frac{t_{w,\max}-t_\infty}{2t_\infty \nu}\, \Delta^2(\eta-\eta^2)\Theta\sin\beta, \tag{7} \]

where \(\eta=y/\Delta\).

With this choice of the function (7), the initial and boundary conditions (5) are satisfied automatically. It is not difficult to see that the function (7) can be obtained in solving equation (3b) if in it one sets \(\partial w_x/\partial \tau=0\), i.e., regards the development of the process as quasi-stationary.

This is quite admissible from the standpoint of physical meaning, since the development in time of the motion process near the body in free convection occurs sufficiently slowly. Equation (7) can be written differently:

\[ w_x = \frac{1}{2}\,Gr\left(\frac{\Delta}{D}\right)^2 \frac{\nu}{D}\, \Theta\sin\beta(\eta-\eta^2), \tag{8} \]

where

\[ Gr = \frac{gD^3}{\nu^2}\cdot \frac{t_{w,\max}-t_\infty}{t_\infty}, \tag{9} \]

or

\[ Re_x=\frac{w_xD}{\nu} = \frac{1}{2}\,Gr\left(\frac{\Delta}{D}\right)^2 \Theta\sin\beta(\eta-\eta^2). \tag{10} \]

Further, equation (3c), taking into account (3a), can be written as

\[ \frac{\partial \Theta}{\partial \tau}+\frac{\partial}{\partial x}(w_x\cdot \Theta) = a\,\frac{\partial^2 \Theta}{\partial y^2}. \tag{11} \]

Or, in dimensionless form,

\[ \frac{\partial \Theta}{\partial Fo} +2\frac{\partial}{\partial \beta}(Pe_x\cdot \Theta) =\frac{\partial^2 \Theta}{\partial y_1^2}, \tag{12} \]

where

\[ Fo=\frac{a\tau}{D^2};\qquad Pe_x=\frac{w_xD}{a},\qquad y_1=\frac{y}{D}. \]

If it is taken into account that \(x=\dfrac{D}{2}\sin\beta\), and \(dx=\dfrac{D}{2}d\beta\) (for small angles \(\sin\beta\approx\beta\)), then, substituting expression (10) into (12), we shall have

\[ \frac{\partial}{\partial Fo} +\frac{\partial}{\partial \beta} \left[ (Gr\cdot Pr)(\eta-\eta^2)\sin\beta \left(\frac{\Delta}{D}\right)^2 \Theta^2 \right] = \frac{\partial^2\Theta}{\partial y_1^2}. \tag{13} \]

The equation obtained is nonlinear with respect to “\(\Theta\)”; consequently, it does not appear possible to solve it by ordinary analytical methods.

Below is the rational linearization of equation (13) proposed by N. L. Men’shikh [2], which approximates the actual result with sufficient accuracy.

We shall approximate

\[ \Theta^2=B\cdot\Theta. \tag{14} \]

We find the coefficient \(B\) from the condition of the minimum of the functional

\[ \delta\int_0^1(\Theta^2-B\Theta)^2\,d\Theta=0, \tag{15} \]

whence we find

\[ B=\frac{3}{4}. \tag{16} \]

Substituting (14) into (13) and taking (16) into account, we shall have

\[ \frac{\partial\Theta}{\partial Fo} +\frac{\partial}{\partial\beta} \left[ \frac{3}{4}(Gr\cdot Pr)(\eta-\eta^2)\sin\beta\cdot\Theta \left(\frac{\Delta}{D}\right)^2 \right] = \frac{\partial^2\Theta}{\partial y_1^2}. \tag{17} \]

The solution of equation (17), under the initial and boundary conditions

\[ Fo=0,\ \Theta=0;\quad \eta=0,\ \Theta=\Theta_w;\quad \eta=1,\ \Theta=0, \tag{18} \]

will be sought in the form of a sum

\[ \Theta=\Theta_1(y_1,\beta)+\Theta_2(Fo,y_1). \tag{19} \]

Upon substitution of (19), equation (17) splits into the following two equations:

\[ \frac{\partial}{\partial\beta} \left[ \frac{3}{4}(Gr\cdot Pr)(\eta-\eta^2) \left(\frac{\Delta}{D}\right)^2 \sin\beta\,\Theta_1 \right] = \frac{\partial^2\Theta_1}{\partial y_1^2}, \tag{20} \]

\[ \frac{\partial\Theta_2}{\partial Fo} = \frac{\partial^2\Theta_2}{\partial y_1^2}. \tag{21} \]

Equation (20) is solved under the following conditions:

\[ Fo \geqslant 0,\quad \eta=0,\quad \Theta_1=1;\quad \eta=1,\quad \Theta_1=0; \tag{22} \]

and equation (21) under the conditions:

\[ \eta=0,\quad \Theta_2=0;\quad \eta=1,\quad \Theta_2=0. \tag{23} \]

Let us integrate both sides of expression (20) and, changing on the right-hand side to the variable \(\eta\), we obtain

\[ \frac{\partial}{\partial \beta} \int_0^1 \left[ \frac{3}{4}(Gr\cdot Pr)(\eta-\eta^2) \left(\frac{\Delta}{D}\right)^2 \sin\beta\cdot \Theta_1 \right]\,d\eta = -\frac{D^2}{\Delta^2} \left. \frac{\partial \Theta_1}{\partial \eta} \right|_{\eta=0}. \tag{24} \]

The boundary conditions (22) are satisfied in the first approximation if for \(\Theta_1\) one chooses the approximating function

\[ \Theta_1=\Theta_w(1-\eta). \tag{25} \]

Substituting (25) into (24), we arrive at the following expression:

\[ \frac{d}{d\beta}\left(\frac{\Delta}{D}\right)^4\sin\beta + \left(\frac{\Delta}{D}\right)^4\cos\beta = \frac{48}{3}\,\frac{1}{Gr\cdot Pr}. \tag{26} \]

Putting \(\beta=0\) in this equation, we find the value of \(\left(\dfrac{\Delta_0}{D}\right)\) at the front critical branching point:

\[ \frac{\Delta_0}{D} = \sqrt[4]{\frac{16}{Gr\cdot Pr}}. \tag{27} \]

Taking (27) into account, equation (26) can be written in the form

\[ \frac{d}{d\beta}\left(\frac{\Delta}{\Delta_0}\right)^4 + \left(\frac{\Delta}{\Delta_0}\right)^4\operatorname{ctg}\beta - \frac{1}{\sin\beta} = 0. \tag{28} \]

The integral of this equation has the form

\[ \left(\frac{\Delta}{\Delta_0}\right)^4 = \frac{1}{\sin\beta}(C+\beta). \tag{29} \]

Since at \(\beta=0\), \(\left(\dfrac{\Delta}{\Delta_0}\right)^4=1\), it follows that \(C=0\).

Consequently, we have

\[ \left(\frac{\Delta}{\Delta_0}\right)^4 = \frac{\beta}{\sin\beta}, \]

whence

\[ \frac{\Delta}{\Delta_0} = \sqrt[4]{\frac{\beta}{\sin\beta}}. \tag{30} \]

To compare the results obtained by using the region of thermal influence with the available experimental data for the stationary regime, we find

\[ Nu_\infty=\frac{D}{\Delta}. \tag{31} \]

Substituting (30) into (31) and taking (27) into account, we shall have

\[ Nu=0.5\sqrt[4]{Gr\cdot Pr}\,\sqrt[4]{\frac{\sin\beta}{\beta}}, \tag{32} \]

and the mean value

\[ \overline{Nu}=\frac{0.5\sqrt[4]{Gr\cdot Pr}}{\pi}\int_{0}^{\pi}\sqrt{\frac{\sin\beta}{\beta}}\,d\beta . \tag{33} \]

Now, regarding the region of influence “\(\Delta\)” as known, we solve equation (21) with boundary conditions (23).

Integrating equation (21), passing at the same time to the variable \(\eta\) on the right-hand side,

\[ \frac{d}{dFo}\int_{0}^{1}\Theta_2\,d\eta = -\left(\frac{D}{\Delta}\right)^2 \left(\frac{\partial \Theta_2}{\partial \eta}\right)_{\eta=0}. \tag{34} \]

Or, averaging over “\(\beta\),” we obtain:

\[ \frac{\partial}{\partial Fo}\int_{0}^{1}\Theta_2\,d\eta = -\left(\overline{\frac{D}{\Delta}}\right)^2 \left(\frac{\partial \Theta_2}{\partial \eta}\right)_{\eta=0}, \tag{35} \]

where

\[ \left(\overline{\frac{D}{\Delta}}\right)^2 = \frac{1}{\pi}\int_{0}^{\pi}\left(\frac{D}{\Delta}\right)^2\,d\beta . \]

Let us express the latter relation, taking into account (30) and (27),

\[ \left(\overline{\frac{D}{\Delta}}\right)^2 = \frac{0.25}{\pi}\sqrt{Gr\cdot Pr} \int_{0}^{\pi}\sqrt{\frac{\sin\beta}{\beta}}\,d\beta . \tag{36} \]

The simplest polynomial of the form

\[ \Theta_2=C(\eta-\eta^2). \tag{37} \]

corresponds to boundary conditions (23). Substituting (37) into (35), we arrive at the differential equation with respect to \(C\):

\[ \frac{dC}{dFo}=-6\delta^2 C, \tag{38} \]

where, for brevity of notation, we have denoted

\[ \delta^2=\left(\overline{\frac{D}{\Delta}}\right)^2 . \tag{39} \]

Integrating, we obtain

\[ C=C_0\exp(-6\delta^2\cdot Fo). \tag{40} \]

Accordingly, substituting (40) into (37), we obtain

\[ \Theta_2=C_0\exp(-6\delta^2\cdot Fo)(\eta-\eta^2). \tag{41} \]

Substituting (25) and (41) into (19), we shall have

\[ \Theta=\Theta_w(1-\eta)+C_0(\eta-\eta^2)\exp(-6\delta^2\cdot Fo). \tag{42} \]

In equation (42) the unknown is still the arbitrary constant \(C_0\). To determine it, we shall use the initial condition:

\[ \text{at } Fo=0,\quad \Theta_w=1,\quad \Theta=0, \]

therefore

\[ 0=(1-\eta)+C_0(\eta-\eta^2). \tag{43} \]

To determine \(C_0\), we use the condition of the minimum of the functional with respect to the initial condition

\[ \delta \int_{0}^{1} \left\{0-\left[(1-\eta)+C_0(\eta-\eta^2)\right]\right\}^{2}\,d\eta=0. \tag{44} \]

Whence we immediately obtain

\[ C_0=-\frac{5}{2}. \]

Consequently, the final solution will have the form

\[ \Theta=\Theta_w(1-\eta)-\frac{5}{2}(\eta-\eta^2)\exp(-6\bar{\delta}^{\,2}Fo). \tag{45} \]

To determine the heat-transfer coefficient, we use the well-known expression:

\[ Nu=-\frac{1}{\Theta_w}\left(\frac{\partial \Theta}{\partial y_1}\right)_{y_1=0}; \]

\[ Nu=-\frac{1}{\Theta_w}\left(\frac{D}{\Delta}\right) \left(\frac{\partial \Theta}{\partial \eta}\right)_{\eta=0} = \frac{1}{\Theta_w} \left[\Theta_w+\frac{5}{2}\exp(-6\bar{\delta}^{\,2}Fo)\right]\frac{D}{\Delta}, \tag{46} \]

\[ Nu=\frac{D}{\Delta} \left[1+\frac{5}{2\Theta_w}\exp(-6\bar{\delta}^{\,2}Fo)\right]. \tag{47} \]

Mean value:

\[ \overline{Nu}= \frac{0.5\sqrt[4]{Gr\cdot Pr}}{\pi} \left(\int_{0}^{\pi}\sqrt{\frac{\sin\beta}{\beta}}\,d\beta\right) \cdot \left[1+\frac{5}{2\Theta_w}\exp(-6\bar{\delta}^{\,2}Fo)\right]. \tag{48} \]

As \(Fo\to\infty\), expression (48) passes into (33). It is interesting to note that, for \(Fo=0\), from (48) we obtain the maximum value of \(\overline{Nu}\), equal to

\[ \overline{Nu}_{\max}=\frac{7}{2}\,Nu_\infty. \tag{49} \]

For subsequent use, equation (48) may be written in the more convenient form

\[ \overline{Nu}=C(Gr\cdot Pr)^{0.25} \left[1+\frac{5}{2\Theta_w}\exp(-6\bar{\delta}^{\,2}Fo)\right] \]

or

\[ \overline{Nu}=Nu_\infty \left[1+\frac{5}{2\Theta_w}\exp(-6\bar{\delta}^{\,2}Fo)\right]. \tag{50} \]

In this expression, the quantity \(\Theta_w\) remains as yet unknown. To find it, we use equation (1a), which we write in dimensionless form

\[ -\frac{d\Theta_w}{dFo}=A\overline{Nu}\,\Theta_w, \tag{51} \]

where

\[ A=4\,\frac{C_{p\infty}\rho_\infty}{C_{pm}\rho_m}. \]

Substituting expression (50) into (51), we arrive at the following equation for \(\Theta_w\):

\[ \frac{d\Theta_w}{dFo}+A Nu_{\infty}\left[\Theta_w+\frac{5}{2}\exp(-6\bar{\delta}^{\,2}Fo)\right]=0, \tag{52} \]

whose integral has the form

\[ \Theta_w=\exp(-ANu_{\infty}\cdot Fo)\left[C_0-\frac{5}{2}\int \exp\{[-6\bar{\delta}^{\,2}-ANu_{\infty}]Fo\}\,dFo\right], \tag{53} \]

whence, using the initial condition according to which, at \(Fo=0\), \(\Theta_w=1\), we finally obtain

\[ \Theta_w=\exp(-ANu_{\infty}Fo)\left|1-\frac{5}{2}\frac{1}{6\bar{\delta}^{\,2}-ANu_{\infty}}\times\right. \]

\[ \left.\times\{1-\exp[-(6\bar{\delta}^{\,2}-ANu_{\infty})Fo]\}\right|. \tag{54} \]

Substituting (54) into (52), we arrive at the computational equation for \(\overline{Nu}\):

\[ \overline{Nu}=\frac{6\bar{\delta}^{\,2}}{A}\left[1+\frac{5}{2\Theta_w}\exp(-6\bar{\delta}^{\,2}Fo)\right]. \]

Conclusions

The introduction of the region of thermal influence made it possible to obtain computational equations for the coefficient of convective heat exchange during the cooling of an infinitely long cylinder by unsteady natural convection. In the limiting case we pass to computational relations corresponding, within the accuracy of the assumptions made, to the known formulas of M. A. Mikheev for the steady regime.

LITERATURE

1. N. S. Koshlyakov, E. B. Gliner, M. M. Smirnov, Basic Differential Equations of Mathematical Physics. Izd. “Nauka,” Moscow, 1959.

2. L. I. Kudryashev, N. L. Menshikh. “The coupled problem of heat exchange in the presence of radiative heat exchange.” Izv. vuzov, “Aviation Technology,” No. 1, 1969.