Heal conduction

Modem installations often impose furnace conditions so severe that refractories other than lire-clay are needed. High aluminum and silicon carbide refractories are typical of tlie.se. The heal conductivities of the super-refractories are larger than those of lire-clay brick, and such construction should be backed up with high temperature insulation. Silicon carbide blocks are the most refractory and have the quality of resisting clinker adhesion heitcr than ordinary fire-brick. Their fusion temperature is about 4000 F (2204 C)... 

Various other physical processes lead in their mathematical description to equations of the same form as Flq (2). especially in its steady-state form, Such processes include the conduction of electricity in a conductor, or the shape of a thin membrane stretched over a curved boundary. This situation has led to the development of analogies (electric analogy, soap film analogy) to heal conduction processes, which are useful because they often offer the advantages of simpler experimentation. 

A.S mentioned in the chapter opener, heat is transferred by the mechanisms of conduction, convection, and radiation, and heat often changes vehicles as it is transferred from one medium to another. For example, the heal conducted to the outer surface of the wall of a house in winter is convected away by the cold outdoor air while being radiated to the cold surroundings. In such cases, it may be necessary to keep track of the energy interactions at the surface, and this is done by applying the conservation of energy principle to the surface. 

The rale of heal conduction through a solid is directly proportional to its thermal conductivity. 

In Solids, heal conduction is due to two effects the lattice vibrational waves induced by the vibrational motions of the molecules po.sitioned at relatively fixed positions in a periodic manner called a lattice, and the energy transported via the free flow of electrons in the solid (Fig. 1—28). The Ihermal conductivity of a solid is obtained by adding the lattice and electronic components. The relatively high thermal conductivities of pure metals arc primarily due to the electronic component. The lattice component of thermal conductivity strongly depends on the way the molecules are arranged. For example, diamond, which is a liighly ordered crystalline solid, has the highest known thermal conductivity at room temperature. 

In Chapter 1 heal conduction was defined as the transfer of thermal energy from the more energetic particles of a medium to the adjacent less energetic ones. It was stated that conduction can take place in liquid.s and gases as well as solids provided that there is no bulk motion involved. 

A medium through which heat is conducted may involve the conversion of mechanical, electrical, nuclear, or chemical energy into heat (or thermal energy). In heal conduction analysis, such conversion processes are characterized as heal (or thermal energy) generation. 

Noting that the area A is constant for a plane wall, the one-dimensional transient heal conduction equation in a plane wall becomes... 

Three-dimensional heal conduction through a rectangular volume element. 

In the last section we considered one-dimensional heat conduction and assumed heat conduction in other directions to be negligible. Most heat transfer problems encountered iu practice can be approximated as being onedimensional, and we mostly deal with such problems in tliis text. However, this is not always the case, and sometimes we need to consider heat transfer in other directions as well. In such cases heal conduction is said to be multidimensional, and in this section we develop the governing differential equation in such systems in rectangular, cylindrical, and spherical coordinate systems. 

The heat conduction equation is first order in time, and thus the initial condition cannot involve any derivatives (it is limited to a specified temperature). However, the heal conduction equation is second order in space coordinates, and thus a boundary condition may involve first derivalives at the boundaries as well as specified values of temperature. Boundary conditions most commonly encountered in practice are the specified temperature, specified heat flux, convection, and radiation boundary conditions. 

Consider a 2-m-hlgh and 0.7-m-wide bronze plate whose thickness is 0.1 m, One side of the plate is maintained at a constant temperature of 600 K vrhile the other side is maintained at 400 K, as shown in Fig. 2-65. The thermal conductivity of the bronze plate can be assumed to vary linearly in that temperature range as k T) = fed + pT) where fe = 38 W/m K and p = 9,21 x lQ-4 ) -i Disregarding the edge effects and assuming steady one-dimensional heat transfer, determine the rate of heal conduction through the plate. 

The heal conduction equation can pe derived by performing an energy balance on a differential volume element. The onc-dimcnsional beat conduction equation in rectangular, cylindjdcal, and spherical coordinate systems for the case of constant thermal conductivities are expressed as... 

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... 

Consider a medium in which the heal conduction equation is given in its simplest form as... 

C What is a boundary condition How many boundary conditions do wc need to specify for a two-dimensional heal conduction problem ... 

Z-Xl Water flows through a pipe at an average temperature of VO C. The inner and outer radii of the pipe are rf = 6 cm and rj = 6.5 cm. respectively. The outer surface of the pipe is wrapped with a thin electric heater that consumes 300 W pet m length of the pipe. The exposed surface of the heater is heavily insulated so that the entire heal generated in the heater is transferred to the pipe. Heat is transferred from the inner surface of the pipe to the water by convection with a heat transfer coefficient of h = 85 W/m K. Assuming constant thermal conductivity and one-dimensionat heat transfer, express the mathematical formulation (the differential equation and the boundary conditions) of the heal conduction in the pipe during steady operation. Do not solve. 

W/m . Assuming steady one-dimensional heat transfer, (a) express the differential equation and the boundary conditions for heal conduction through the wall, (b) obtain a relation for the variation of temperature in the wall by solving the differential... 

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. 

Consider a 20-cm-ihick large concrete plane wall k 0.77 V/in °C) subjected to convection on both sides with r, = 27"C and A, = 5 W/m °C on the inside, and = 8°C and A2 = 12 W/m °C on the outside. Assuming constant thermal conductivity with no heat generation and negligible radiation, [a) express the differential equations and the boundary conditions for steady one-dimensional heal conduction through the wall, (A) obtain a relation for the variation of temperature in the wall by solving the differential equation, and (c) evaluate the temperatures at the inner and outer surfaces of the wall. 

Steady one-dimensional heal conduction takes place in a long slab of width U (In the direction of heat flow, x) and thickness Z. The slab s thermal conductivity varies with temperature ask = k /(T -t- T), where Tis the temperature (in K), and k (in W/m) and T (in K) are two constants. The temperatures at X = 0 and x IV ate To and T,, re.spectively. Show that the heat flux in steady operation is given by... 

The heal conduction equation in cylindrical or spherical coordinates can be nondimensiooalized in a similar way. Note that nondimeosionalization... 

The analytical solution obtained above for one-dimensional transient heal conduction in a plane wall involves infinite series and implicit equations, which aris.difficult to evaluate. Therefore, there is clear motivation to simplify... 

Fourier number at time / can be viewed as the ratio of the rate of heal conducted to the rate of heat stored at that time. 

The transient temperature charts and analytical solutions presented earlier can be used to determine the temperature distribution and heat transfer in one-dimensioual heal conduction problems associated with a large plane wall, a... 

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 solutions of transient heal conduction in a semi-infinite solid with constant properties under various boundary conditions at the surface are given as follows ... 

S. Kakaf and Y. Yener, Heal Conduction, New York Hemisphere Publishing Co., 1985. 

To obtain a general difference equation for the interior nodes, consider the element represented by node m and die two neighboring nodes m - I and in f I. Assuming the heal conduction to be into the element on all surfaces, an energy balance on the element can be expressed as... 

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