The Plane Wall

First consider the plane wall where a direct application of Fourier s law [Eq. (1-1)] may be made. Integration yields 

If more than one material is present, as in the multilayer wall shown in Fig. 2-1, the analysis would proceed as follows The temperature gradients in the three materials are shown, and the heat flow may be written 27 

Note that the heat flow must be the same through all sections. 

Solving these three equations simultaneously, the heat flow is written 

At this point we retrace our development slightly to introduce a different conceptual viewpoint for Fourier s law. The heat-transfer rate may be considered as a flow, and the combination of thermal conductivity, thickness of material, and area as a resistance to this flow. The temperature is the potential, or driving, function for the heat flow, and the Fourier equation may be written 

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The Plane Wall. To calculate the heat-transfer rate through a plane wall, Fourier s law can be appHed directly. 

When the plane wall is made of mote than one material (Fig. lb), the heat-transfer rate is given as... 

Let X be the ratio of the cylinder radius a to the distance between the long axis of the cylinder and the plane wall d. For the cylinder migrating perpendicular to the plane, the electrophoretic velocity is expressed by... 

Consider the plane wall shown in Fig. 2-5 exposed to a hot fluid A on one side and a cooler fluid B on the other side. The heat transfer is expressed by... 

Consider the plane wall with uniformly distributed heat sources shown in Fig. 2-8. The thickness of the wall in the x direction is 2L, and it is assumed that the dimensions in the other directions are sufficiently large that the heat flow may be considered as one-dimensional. The heat generated per unit volume is q, and we assume that the thermal conductivity does not vary with temperature. This situation might be produced in a practical situation by passing a current through an electrically conducting material. From Chap. 1, the differential equation which governs the heat flow is... 

The plane wall shown has internal heat generation of 50 MW/tn and thermal properties of k = 19 W/m °C, p = 7800 kg/mJ, and c = 460 J/kg °C. It is initially at a uniform temperature of 100°C and is suddenly subjected to the heat generation and the convective boundary conditions indicated in the figure. Calculate the temperature distribution after several time increments. 

We have already discussed the overall heat-transfer coefficient in Sec. 2-4 with the heat transfer through the plane wall of Fig. 10-1 expressed as... 

From the standpoint of heat-exchanger design the plane wall is of infrequent application a more important case for consideration would be that of a doublepipe heat exchanger, as shown in Fig. 10-2. In this application one fluid flows on the inside of the smaller tube while the other fluid flows in the annular space between the two tubes. The convection coefficients are calculated by the methods described in previous chapters, and the overall heat transfer is obtained from the thermal network of Fig. 10-2h as... 

An examination of the one dimensional tiansient heat conduction equations for the plane wall, cylinder, and sphere reveals that all tlu-ee equations can be expressed in a compact form as... 

Now consider steady one-dimensional heat transfer through a cylindrical or spherical layer that is exposed to convection on boili sides to fluids at temperatures and T 2 with heat transfer coefftcients /t, and h, respectively, as shown in Fig. 3-25. The thermal resistance network in this case consists of one conduction and two convection resistances in series, just like the one for the plane wall, and the rate of heat transfer under steady conditions can be expressed as... 

Consider a plane wall of thickness 2L, a long cylinder of radius r , and a sphere of radius r, initially at a nnifonn temperature T,-, as shown in Fig. 4—11. At time t = 0, each geometry is placed in a large medium that is at a constant temperature T and kepi in that medium for t > 0. Heat transfer lakes place between these bodies and their environments by convection with a uniform and constant heal transfer coefficient A. Note that all three ca.ses possess geometric and thermal symmetry the plane wall is symmetric about its center plane (,v = 0), the cylinder is symmetric about its centerline (r = 0), and the sphere is symmetric about its center point (r = 0). We neglect radiation heat transfer between these bodies and their surrounding surfaces, or incorporate the radiation effect into the convection heat transfer coefficient A. 

The variation of tiie temperature profile with time in the plane wall i.s illustrated in Fig. 4-12. When the wall is first expo.sed to the surrounding medium at 7L < T,- at / 0, the entire wall is at its initial temperature Tj. But the wall temperature at and near the surfaces starts to drop as a result of heal transfer from the wall to the surrounding medium. This creates a temperature... 

This completes the analysts for the solution of one-dimensional transient heat conduction problem in a plane wall. Solutions in other geometries such as a long cylinder and a sphere can be determined using the same approach. The results for all three geometries arc summarized in Table 4—1. The solution for the plane wall is also applicable for a plane wall of thickness L whose left surface at, r = 0 is insulated and the right surface at.t = T. is subjected to convection since this is precisely the mathematical problem we solved. 

The dimensionless quantities defined above for a plane wall can also be used for a cylinder or sphere by replacing the space variable x by r and the half-thickness L by the outer radius r. Note that the characieristic length in the definition of the Biot number is taken to be the half-thickness L for the plane wall, and the radius for the long cylinder and sphere instead of 1//A used in lumped system analysis. 

Using the appropriate nonditnensional temperature relations based on the one-term approximation for the plane wall, cylinder, and sphere, and performing the indicatetl integrations, we obtain the following relations for the fraction of heat transfer in those geometries ... 

Then WG determine the dimensionless heat transfer ratios for bolh geometries. For the plane wall, it Is determined from Fig. 4-15c to be... 

The choice of the plus or minus sign in (5-97) determines whether the unit normal points into the slider block or points into the gap and is the outer normal as required. It is easy to see which sign to choose by considering the limiting case m = 0 in which the surface of the slider block is parallel to the plane wall. In this case... 

If F is constant, it follows from (5-119) that db /dt must decrease at just the rate to balance the increase in h 1 as the gap width decreases. This means that the sphere velocity rapidly decreases as the sphere approaches the plane wall. 

The flow configuration is sketched in the figure. The coordinate direction normal to the plane walls is specified as X3, and the velocity of the center of mass is denoted as Ify. Further, the undisturbed velocity, pressure, and stress are denoted, respectively, as V, Q, T. In the most general case,... 

If we consider a one-dimensional heat flow along the x direction in the plane wall shown in Fig. 1.1a, direct application of Eq. 1.1 can be made, and then integration yields... 

The distribution of the velocities computed at various locations of the plane wall-jet show that the self-similarity is satisfied in the fully developed zone, i.e. x/h > 10. In Figure 3, besides the Verhoff s equation, the Gaussian distribution and the experimental results of Forthmann (from Rajaratnam, 1976) are also shown for comparison. Excepting for y/b j < 0.16, the Gaussian distribution seems to be a good fit to the actual distribution of the normalized velocity. 

The variation of the skin-friction coefficient, Cf = Xg/(0.5pUg ), for the plane wall jet is shown in Figure 6. There seems to be a fair agreement between the predicted values and the ones given by Myer s expression (Rajaratnam, 1976). The measured values of the current four runs exhibit a significant amount of scatter and are not shown here. 

In many practical situations the surface temperatures (or boundary conditions at the surface) are not known, but there is a fluid on both sides of the solid surfaces. Consider the plane wall in Fig. 4.3-3a with a hot fluid at temperature T, on the inside surface and a cold fluid at T, on the outside surface. The outside convective coefficient is/io W/m -K and /i on the inside. (Methods to predict the convective h will be given later in Section 4.4 of this chapter.)... 

Fig. 13 Pressure loading on the plane wall at two fixed times.

To explore the pathway for wall-induced crystallization, we performed Monte Carlo simulations in the constant normal-pressure NPx T) ensemble. Here N refers to the number of hard-spheres in the system. The simulation box was rectangular with periodic boundary conditions in the x and y directions. In the z-direction, the system is confined by two flat, hard walls at a distance L. is the component of the pressure tensor perpendicular to the plane wall, and T is the temperature. As our unit of length we used the hard-sphere diameter a. T only sets the energy scale. In the following we always use reduced units. The state of the bulk hard-sphere system is completely specified by its volume fraction cf). The coexistence volume fractions for the bulk fluid and solid phase are known [27] [Pg.193]

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