Transient heat conduction cylinders

Now consider a sphere, with density p, specific heat c, and outer radius R. The area of the sphere normal to the direclion of heat transfer at any location is A — 4vrr where r is the value of the radius at that location. Note that the heat transfer area A depends on r in this case also, and thus it varies with location. By considering a thin spherical shell element of thickness Ar and repeating tile approach described above for the cylinder by using A = 4 rrr instead of A = InrrL, the one-dimensional transient heat conduction equation for a sphere is determined to be (Fig. 2-17)... 

Starting with an energy balance on a disk volume element, derive the one-dimensional transient heat conduction equation for T(t, r) in a cylinder of diameter D with an insulated side surface for the case of constant thensal conductivity with heat generation. 

We start this chapter with the analysis of lumped systems in which the temperature of a body varies with time but remains uniform throughout at any time. Then we consider the variation of temperature with time as well as position for one-dimensional heat conduction problems such as those associated with a large plane wall, a long cylinder, a sphere, and a semi infinite medium using transient temperature charts and analytical solutions. Finally, we consider transient heat conduction in multidimensional systems by utilizing the product solution. 

TRANSIENT HEAT CONDUCTION IN LARGE PLANE WALLS, LONG CYLINDERS. AND SPHERES WITH SPATIAL EFFECTS... 

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 three bodies — plate, very long cylinder and sphere — shall have a constant initial temperature d0 at time t = 0. For t > 0 the surface of the body is brought into contact with a fluid whose temperature ds d0 is constant with time. Heat is then transferred between the body and the fluid. If s < i90, the body is cooled and if i9s > -i90 it is heated. This transient heat conduction process runs until the body assumes the temperature i9s of the fluid. This is the steady end-state. The heat transfer coefficient a is assumed to be equal on both sides of the plate, and for the cylinder or sphere it is constant over the whole of the surface in contact with the fluid. It is independent of time for all three cases. If only half of the plate is considered, the heat conduction problem corresponds to the unidirectional heating or cooling of a plate whose other surface is insulated (adiabatic). 

Here the Laplace operator V2t has the form given in 2.1.2 for cartesian and cylindrical coordinate systems. We will once again consider the transient heat conduction problem solved for the plate, the infinitely long cylinder and the sphere in section 2.3.4 A body with a constant initial temperature is brought into contact with a fluid of constant temperature tfy so that heat transfer takes place between the fluid and the body, whereby the constant heat transfer coefficient a is decisive. 

Noting that the heat transfer area in this case is A - ItitL, the one-dimensional transient heal conduction equation in a cylinder becomes... 

C Write down the one-dimensional transient beat conduction equation for a long cylinder with constant thermal conductivity and heat generation, and indicate what each variable represents,... 

Based on this relation, determine (heat conduction is steady or transient, (b) if it is one-, two-, or three-dinieiisional, and (c) the value of heat flux on the side. surface of the cylinder at r = r. ... 

The formulation of heat conduction problems for the determination of the one-dimj .nsional transient temperature distribution in a plane wall, a cylinder, or a sphefeTesults in a partial differential equation whose solution typically involves irtfinite series and transcendental equations, wliicli are inconvenient to use. Bijt the analytical soluliop provides valuable insight to the physical problem, hnd thus it is important to go through the steps involved. Below we demonstrate the solution procedure for the case of plane wall. 

Coefficients used in the one-term approximate solution of transient one-dimensional heat conduction in plane walls, cylinders, and spheres (B = hUk for a plane wall of thickness ZL, and Bi = hrjkfor a cylinder or sphere of radius r )... 

Consider a sltori cylinder of height a and radius r initially at a uniform temperature T,. There is no heat generation in the cylinder. At time t = 0. the cylinder is subjected to convection from all surfaces to a medium at temperature l with a heal transfer coefficieiu h. The temperature within the cylinder will change with a as well as r and time f since heal transfer occurs from Ihe top and bottom of the cylinder as weU as its side surfaces. That is, T = 7 (r,, v, f) and thus this is a two-dimensional transient heal conduction problem. When the properties are assumed to be constant, it can be shown that the solution of this two-dimensional problem can be expressed as... 

The diffusion coefficients in solids are typically very low (on the order of 10 to 10" mVs), and thus the diffusion process usually affects a thin layer at the surface. A solid can conveniently be treated as a semi-infinite medium during transient mass diffusion regardless of its size and shape when the penetration depth is small relative to the thickness of the solid. When this is not the case, solutions for one dimensional transient mass diffusion through a plane wall, cylinder, and sphere can be obtained from the solution.s of analogous heat conduction problems using the Heisler charts or one term solutions pieseiited in Chapter 4. 

In section 2.5.3 it was shown that the differential equation for transient mass diffusion is of the same type as the heat conduction equation, a result of which is that many mass diffusion problems can be traced back to the corresponding heat conduction problem. We wish to discuss this in detail for transient diffusion in a semi-infinite solid and in the simple bodies like plates, cylinders and spheres. 

Derive the differential equation for the temperature field = (r, t), that appears in a cylinder in transient, geometric one-dimensional heat conduction in the radial direction. Start with the energy balance for a hollow cylinder of internal radius r and thickness Ar and execute this to the limit Ar — 0. The material properties A and c depend on internal heat sources are not present. 

Now we consider application of the transform technique to a transient problem of heat conduction or Fickian diffusion in a cylinder (Fig. 11.4). The slab problem (Example 11.1) was dealt with in Section 11.2.2. 

Tests at Babcock and Wilcox (B W) (Reference 22) simulated the approximate flow decay and power transient in a single full-size, electrically-heated fuel assembly mockup. The FIs were approximately 10 percent higher than those predicted by FLOWTRAN-FI using OSV as a predictor of FI. While the coolant flow recovered after about 0.1 seconds in the B W tests, due to system hydraulic "bounce-back," there was generally no recovery from FI once it occurred. The heater cylinders in the B W tests were Monel, as opposed to an aluminum alloy used for the SRS fuel cylinders. The good conductivity of the SRS fuel might possibly provide some margin to recovery from FI. 

A spherical metal ball of radius is heated in an oven to a temperature of T, throughout and is then taken out of the oven and allowed to cool in ambient air at T by convection and radiation. The emissivily of the outer surface of (he cylinder is c, and the temperature of the surrounding surfaces is The average convection heat transfer coefficient is estimated to be II Assuming variable thermal conductivity and transient one-diiiiensional lieat transfer, express the mathematical formulation (the differential equation and the boundary... 

In the case of the thermal-conductivity, there are three main techniques those based on Equation (1) and those based on a transient application of it. Prior to about 1975, two forms of steady-state technique dominated the field parallel-plate devices, in which the temperature difference between two parallel disks either side of a fluid is measured when heat is generated in one plate, and concentric cylinder devices that apply the same technique in an obviously different geometry. In both cases, early work ignored the effects of convection. In more recent work, exemplified by the careful work in Amsterdam with parallel plates, and in Paris with concentric cylinders, the effects of convection have been investigated. Indeed, the parallel-plate cells employed in Amsterdam by van den Berg and his co-workers have the unique feature that, because the temperature difference imposed can be very small and the horizontal fluid layer very thin, it is possible to approach the critical point in a fluid or fluid mixture very closely (mK). 

A newer method is the transient hot-wire method, where an electric current is passed through a metal wire immersed in the fluid. The resistance of the wire is affected by its temperature, which in turn is affected by the dissipation of heat from the wire s surface, which depends on the thermal conductivity of the fluid. These instruments require sophisticated data analysis, but that is no longer an obstacle with the ready availability of personal computers. The absence of convection is relatively easy to verify. The best research instruments can achieve an accuracy of better than 1%. Measurements on conducting fluids (such as polar liquids) are more difficult because of the need to electrically insulate the wire. Other geometries, such as needle-shaped cylinders and thin strips, are also sometimes used for transient measurements. 

The analytical solutions for transient conduction in plates, cylinders, and spheres have been obtained by Heisler [9] and the solutions represented graphically for more convenient use. These solutions are for the case of a solid of initially uniform temperature Tg exposed at time zero to a surrounding fluid medium at a constant temperature T. The surface of the solid is cooled or heated by the fluid with a constant convective heat-transfer coefficient h. The sohd is assumed to have a constant thermal conductivity and a constant thermal diffusivity a, defined as... 

The essential difference between a steady state and a transient state is that the temperature at a particular location changes with time under transient conditions. A line heat source probe has been recommended by many researchers [28,29,52,53]. The method is simple, fast, and requires a relatively small sample. A schematic representation of the thermal conductivity probe, the direct current (dc) supply, and the temperature measuring system is shown in Figure 24.5 [54]. The probe is inserted into a sample of a uniform temperature and is heated at a constant rate. The temperature adjacent to the line heat source is recorded. Various modifications of the line heat source probe can be found in the literature. The probe attached to a 20-cm diameter aluminum cylinder as a sample holder is one of them (Figure 24.6) [35]. Other modifications are related to placement of thermocouples directly on the heating element [55,56] or at a fixed distance from it [43]. 

Equation 7.48 with boundary conditions 7.49 is simply the equation for transient conductive heat transfer in a cylinder, where z is the timelike variable, except that Bi varies with z. We know that Bi is relatively insensitive to the velocity and can be approximated by a constant value in the lower portion of the spudine, where little attenuation occurs. The solution to Equations 7.48 and 7.49 for constant Bi is... 

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