Cylinders unsteady state

Fig. 5.6 Temperature profiles for unsteady-state heat conduction in infinite cylinders r(r,0) = To,T(R,t) = T. [Reprinted by permission from H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed., Oxford University Press, New York, 1973.]...

Step 4 Develop the unsteady-state balance(s) for the process. In an elemental time At, the height of the water in the tank drops A/i. The mass of water leaving the tank is in the form of a cylinder 5 cm in area, and we can calculate the quantity as... 

General solutions of unsteady-state conduction equations are available for certain simple shapes such as the infinite slab, the infinitely long cylinder, and the sphere. For example, the integration of Eq. (10.16) for the heating or cooling of an infinite slab of known thickness from both sides by a medium at constant... 

Average temperatures during unsteady-state heating or cooling of a large slab, an infinitely long cylinder, or a sphere. 

Divisions of Analysis. The preceding model describes conditions within a single fetal capillary surrounded by a thin tissue cylinder and supplied by a cylindrical annulus of maternal blood, as shown in Figure 3. Since the numerical techniques required for the solution of such equations were not well defined, the determination of a steady-state concurrent solution was first obtained. Based upon the results of this work, an unsteady-state concurrent solution was assumed possible and feasible. 

After regarded U-vertical pipe as an equivalent pipe, soil temperature field which around it is a cylinder temperature field, in the circular direction there is no temperature gradient, the temperature distribution of the concrete around can be regarded as an axisymmetric problems. The styles of differential equations of heat conductivity are axisymmetric and unsteady state ... 

The numerical results of this case are presented graphically in Figs. 5.3-5 and 5.3-6. Figure 5.3-5 by Gurney and Lurie (G2) is a convenient chart for determining the temperatures at any position in the plate and at any time t. The dimensionless parameters used in these and subsequent unsteady-state charts in this section are given in Table 5.3-1 (x is the distance from the center of the flat plate, cylinder, orsphere. x, is one half the thickness of the flat plate radius of cylinder, or radius of sphere, x = distance from the surface for a semiinfinite solid.)... 

Here we consider unsteady-state conduction in a long cylinder where conduction occurs only in the radial direction. The cylinder is long so that conduction at the ends can be neglected or the ends are insulated. Charts for this case are presented in Fig. 5.3-7 for determining the temperatures at any position and Fig. 5.3-8 for the center temperature only. 

FIGURE 5.3-7. Unsteady-state heat conduction in a long cylinder. [ From H. P. Gurney and J. Lurie, Ind. Eng. Chem., 15,1170 1923). ]... 

Figuri- 5.3-8. Chart for determining temperature at the center of a long cylinder for unsteady-state heat conduction. [From H. P. Hetsler, Trans. A.S.M.E., 69,227 (1947). With permission.]... 

If the surface resistance is negligible, the curves given in Fig. 5.3-13 will give the total fraction of unaccomplished change, E, for slabs, cylinders, or spheres for unsteady-state... 

Unsteady-state conduction in a cylinder. In deriving the numerical equations for unsteady-state conduction in a flat slab, the cross-sectional area was constant throughout. In a cylinder it changes radially. To derive the equation for a cylinder. Fig. 5.4-3 is used where the cylinder is divided into concentric hollow cylinders whose walls are Ax m thick. Assuming a cylinder 1 m long and making a heat balance on the slab at point n, the rate of heat in — rate of heat out = rate of heat accumulation. 

Unsteady-State Conduction in a Short Cylinder. An aluminum cylinder is initially... 

Unsteady-State Diffusion in a Cylinder of Agar Gel. A wet cylinder of agar gel at... 

In many cases the temperature of a process that is varying continuously with time is determined experimentally by measuring the temperature in the slowest-heating region. In cans this is the center of the can. Methods given in Chapter 5 for unsteady-state heating of short, fat cylinders by conduction can be used to predict the center temperature of the can as a function of time. However, these predictions can be somewhat in error, since physical and thermal properties of foods are difficult to measure accurately and often can vary. Also, trapped air in the container and unknown convection effects can affect the accuracy of predictions. 

In the steady-state permeation method, the surfaces of a polymer film of thickness i = 2b are kept at constant gas pressures Pao an PAr- At steady state Eq. 4.75 applies, and the permeability P is found from this equation. If the solubility is known, then the diffusivity can be found from Eq. 4.71. For hollow cylinders and spheres, expressions similar to Eq. 4.74 hold. On the other hand, in the time lag method we deal with the unsteady-state permeation of a diffusant through a slab of thickness 2b. The surfaces of the slab are kept at concentrations Cao and at zero. The accumulated amount of gas which has passed through the slab in time t. 

This variable is the argument of the error function of the semi-infinite slab, it determines the standard deviation of the decaying pulse, and it is central to the time dependence of diffusion into the cylinder. In other words, it is a key to all the foregoing unsteady-state problems. Indeed, it can be easily isolated by dimensional analysis. 

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