Steady heat conduction cylinders

FIGURE 2-16 Tivo equivalent forms of the differential equation for the one-dimensional steady heat conduction in a cylinder with no heat generation. 

We stait this chapter with one-dimensional steady heat conduction in a plane wall, a cylinder, and a sphere, and develop relations for thennal resistances in these geometries. We also develop thermal resistance relations for convection and radiation conditions at the boundaries. Wc apply this concept to heat conduction problems in multilayer plane wails, cylinders, and spheres and generalize it to systems that involve heat transfer in two or three dimensions. We also discuss the thermal contact resislance and the overall heat transfer coefficient and develop relations for the critical radius of insulation for a cylinder and a sphere. Finally, we discuss steady heat transfer from finned surfaces and some complex geometries commonly encountered in practice through the use of conduction shape factors. 

For one-dimensional steady-state heat conduction with no heat generation, in a cylinder of length H, if the first kind boundary conditions... 

Conduction with Heat Source Application of the law of conservation of energy to a one-dimensional solid, with the heat flux given by (5-1) and volumetric source term S (W/m3), results in the following equations for steady-state conduction in a flat plate of thickness 2R (b = 1), a cylinder of diameter 2R (b = 2), and a sphere of diameter 2R (b = 3). The parameter b is a measure of the curvature. The thermal conductivity is constant, and there is convection at the surface, with heat-transfer coefficient h and fluid temperature I. ... 

We start this chapter with a description of steady, unsteady, and multidimensional heat conduction. Then we derive the differential equation that governs heat conduction in a large plane wall, a long cylinder, and a sphere, and generalize the results to three-dimensional cases in rectangular, cylindrical, and spherical coordinates. Following a discussion of the boundary conditions, we present tlie formulation of heat conduction problems and their solutions. Finally, we consider lieat conduction problems with variable thermal conductivity. 

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. ... 

Consider a short cyUnder of radius r<, and height H in which heat is generated at a constant rate of Heat is lost from the cylindrical surface at r = r by convection to the surrounding medium at temperature with a heat transfer coefficient of /i. The bottom surface of the cylinder at z = 0 is insulated, while the top surface at z — is subjected to uniform heat flux Assuming constant thermal conductivity and steady two-dimensional heat transfer, express the mathematical formulation (the differential equation and the boundary conditions) of this heat conduction problem. Do not. solve. 

Which one of the followings is the correct expression for one-dimensional, steady-stale, constant thermal conductivity heat conduction equation for a cylinder with heat generation ... 

As a simple, but practically important application, the conduction of heat independent of time, so called steady conduction, in a flat plate, in a hollow cylinder and in a hollow sphere will be considered in this section. The assumption is made that heat flows in only one direction, perpendicular to the plate surface, and radially in the cylinder and sphere, Fig. 1.3. The temperature field is then only dependent on one geometrical coordinate. This is known as one-dimensional heat conduction. 

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). 

Because the empty cylinder of FGM is placed in steady state temperature field, two ends are heat insulation, the temperature distribution T is no relation with z, 6 and time t. One dimensional equation of heat conduction in r direction is given as ... 

In this work, heat and fluid flow in some common micro geometries is analyzed analytically. At first, forced convection is examined for three different geometries microtube, microchannel between two parallel plates and microannulus between two concentric cylinders. Constant wall heat flux boundary condition is assumed. Then mixed convection in a vertical parallel-plate microchannel with symmetric wall heat fluxes is investigated. Steady and laminar internal flow of a Newtonian is analyzed. Steady, laminar flow having constant properties (i.e. the thermal conductivity and the thermal diffusivity of the fluid are considered to be independent of temperature) is considered. The axial heat conduction in the fluid and in the wall is assumed to be negligible. In this study, the usual continuum approach is coupled with the two main characteristics of the microscale phenomena, the velocity slip and the temperature jump. 

Consider one-dimensional steady-state heat conduction in a cylinder with internal heat generation and convection boundary condition. 

The steady-state method is also known as divided bar method (see, for example. Beck, 1957 Chekhonin et al., 2012). A cylindrical rock sample is positioned (sandwiched) between two cylinders composed of a reference material with known thermal conductivity. This is a thermal series system. The end of one reference cylinder is heated. After steady state is reached, the temperature drop within the rock sample is compared with the temperature drop within the reference cylinders. The comparison gives thermal conductivity of the rock sample. 

Question Derive an expression analogons to Eq. (4.54) for steady-state heat flow through concentric cylinders, as shown at right. All cylinders are of the same length, L. Assume that thermal conductivity is constant with temperature, but that the material in each concentric ring is different. 

Starting with an energy balance on a cylindrical shell volume element, derive the steady one-dimensional heal conduction equation fora long cylinder with constant tliemial conductivity in which heat is generated at a rate of... 

C Consider steady oac-dimeosional heal conduction in a plane wall, long cylinder, and sphere with constant thermal conductivity and no heat generation. Will the temperature in any of these mediums vary linearly Explain. 

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). 

The traditional way to measure thermal conductivity is with steady-state instruments, in which a measured heat flux is compared to a temperature difference between surfaces. Most often the geometry is coaxial cylinders, a thin wire inside a cylinder, or parallel plates. In such instruments, eliminating convection currents is crucial many old data taken with steady-state instruments are unreliable because of convection. Multiple experiments at different heat fluxes are often performed to verify the absence of convection. With good design and operation, such instruments may achieve accuracy in the 1% to 3% range. 

Let the rate of energy per unit volume generated in a solid cylinder or a solid sphere be u" (r) = u , the radius and the thermal conductivity of the cylinder or the sphere be R and k(T) = kr (alternate notations ur and kr are used for convenience in the following formulation). Under steady conditions, the total energy generated in the cylinder or sphere is transferred, with a heat transfer coefficient h, to an ambient at temperature Too. This cylinder could be one of the fuel rods of a reactor core, or one of the elements of an electric heater, and the cylinder or sphere could be a bare, homogeneous reactor core. We wish to determine the radial temperature distribution. 

EXAMPLE 5.6-1. Temperature Profile with Heat Generation A solid cylinder in which heat generation is occurring uniformly asq W/m is insulated on the ends. The temperature of the surface of the cylinder is held constant at K. The radius of the cylinder s r = R m. Heat flows only in the radial direction. Derive the equation for the temperature profile at steady state if the solid has a constant thermal conductivity. 

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]. 

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