Heat Conduction in a Cylinder

Consider heat conduction in a cylinder. [2] The dimensionless temperature profile is governed by 

Since BesselY becomes infinite x = 0 C2 should be zero and the solution is taken as  

8 Laplace Transform Technique for ParaboUc Partial Differential Equations for Time Dependent Boundary Conditions - Use of Convolution Theorem 

In example 8.9 the Laplace transform technique was used to solve a time dependent problem. Inversing the Laplace transform is not straightforward. For complicated time dependent boundary conditions the convolution theorem can be used to find the inverse Laplace transform efficiently. If H(s) is the solution obtained in the Laplace domain, H(s) is represented as a product of two functions  

8 Laplace Transform Technique for Partial Differential Equations 

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

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

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

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. 

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. 

Analytical solution methods arc Limited to highly simplified problems in simple geomeiries (Fig. 5-2). The geometry must be such that its entire surface can be described mathematically in a coordinate system by setting the variables equal to constants. That is, it must fit into a coordinate system perfectly with nothing sticking out or in. In the case of one-dimensional heat conduction in a solid sphere of radius r, for example, the entire outer surface can be described by r - Likewise, the surfaces of a finite solid cylinder of radius r and height H can be described by r = for the side surface and z = 0 and... 

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). ]... 

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. 

Heat Conduction in a Solid Circular Cylinder with Heat Generation—Fuel Rod.. 

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... 

The problem of heat conduction in a semicircular cylinder with one cavity subject to convection boundary conduction is obtained in this investigation ... 

A liquid solution may be separated into its constituents by crystallising out either pure solvent or pure solute, the latter process occurring only with saturated solutions. (At one special temperature, called the cryohydric temperature, both solvent and solute crystallise out side by side in unchanging proportions.) We now consider what happens when a small quantity of solute is separated from or taken up by the saturated solution by reversible processes. Let the saturated solution, with excess of solute, be placed in a cylinder closed below by a semipermeable septum, and the w7hole immersed in pure solvent. The system is in equilibrium if a pressure P, equal to the osmotic pressure of the saturated solution when the free surface of the pure solvent is under atmospheric pressure, is applied to the solution. Dissolution or precipitation of solute can now be brought about by an infinitesimal decrease or increase of the external pressure, and the processes are therefore reversible. If the infinitesimal pressure difference is maintained, and the process conducted so slowly that all changes are isothermal, the heat absorbed when a mol of solute passes into a solution kept always infinitely... 

In the case K > fi, the usual diffusion determines the kinetics for any gel shapes. Here the deviation of the stress tensor is nearly equal to — K(V u)8ij since the shear stress is small, so that V u should be held at a constant at the boundary from the zero osmotic pressure condition. Because -u obeys the diffusion equation (4.18), the problem is trivially reduced to that of heat conduction under a constant boundary temperature. The slowest relaxation rate fi0 is hence n2D/R2 for spheres with radius R, 6D/R2 for cylinders with radius R (see the sentences below Eq. (6.49)), and n2D/L2 for disks with thickness L. However, in the case K < [i, the process is more intriguing, where the macroscopic critical mode slows down as exp(- Q0t) with Q0 oc K. 

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. ... 

A careful test of Nernst s theory was reported by Isnardi243 in 1915. His work consisted of the measurement of the thermal conductivity of iodine (whose dissociation as a function of temperature and pressure was well known), comparison of the experimental results with those of Nernst s theory, and the use of the method to determine JD(H2) which was not then known. Isnardi measured thermal conductivities by the hot wire method, in which the rate of heat removal by conduction from the wire arranged concentrically in a cylinder containing the gas is given by... 

One-dimensional heat conduction through a volume element in a long cylinder,... 

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

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. 

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. 

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. 

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... 

SOLUTION A short cylinder is allowed to cool iri atmospheric air. The temperatures at the centers of the cylinder and the top surface are to be determined, Assumptions 1 Heat conduction In the short cylinder is two dimensional, and thus the temperature varies in both the axial x- and the radial r-directions. 2 The thermal properties of the cylinder and the heat transfer coefficient are constant. 3 Tho Fourier number is t > 0.2 so that the one-teriri approximate solutions are applicable. 

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. 

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. 

Consider the heat conduction/mass transfer problem in a cylinder.[6] [9] [10] The governing equation in dimensionless form is... 

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