Heal conduction equations

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

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. 

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... 

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

It can be shown that the differential equations for both heat conduction and mass diffusion are of the same form. Therefore, the solutions of mass diffusion equations can be obtained from Ihe solutions of corresponding heal conduction equations for the same type of boundary conditions by simply switching the corresponding coefficienis and variables. 

A Euler-Lagrange equation for the variational problem of sj L dv may be obtained by considering the differential heal conduction equation, and wc have... 

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. 

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. 

Starting with an energy balance on a spherical shell volume clement, derive the one-dimensional transient heat conduction equation for a sphere with constant thermal conductivity and no heal generation. 

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... 

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. 

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... 

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... 

TliCf riiiite difl erence fomiulatioii of steady heal conduction problems usu ally results in a system of iV algebraic equations in /V unknown nodal temperatures that need to be solved siiiiullaneously. When Af is small (such as 2 or 3), we can use the elementary elimination method to eliminate ail unknowns except one and then solve for that unknown (sec Example 5-1). The other unknowns are then determined by back substitution. When W is large, which is usually Uie case, the elimination luelliod is not practical and we need to use a more systematic approach that can be adapted to computers. 

A number of problems such as the diying of porous solids of various shapes fell into this category. Solutions to such similar problems may be found in Crank, Mathematics of Dijfiision, and also in Catslaw and Jaeger, Heal Conduction in Solids, since the unsteady-state diffusirm equation often takes on the same form as that for heat conduction. 

The pipe is then lagged with a 50 mm thickness of lagging of thermal conductivity 0.1 W/m K. If the outside heat transfer coefficient is given by the same equation as for the bare pipe, by what factor is the heal loss reduced ... 

Which equation below is used lo determine the heal flux for conduction ... 

B Understand mullidimensionality and time dependence of heat transfer, and the conditions under which a heal transfer problem can be approximated as being one-dirnensional, B Obtain the differential equation of heat conduction in various coordinate systems, and simplify it for steady one-dimensional case,... 

The thermal conductivity is given to be constant, and there is no heal generation in the billet. Therefore, the differential equation that governs the variation... 

Consider a small hot metal object of mass m and specific heal c lliat is initially at a temperature of 7j. Now the object is allowed to cool in an environment at T by convection with a heat transfer coefficient of /r. The (emperanire of the metal object is observed to vary uniformly with time during cooling. Writing an energy balance on the entire metal object, derive the differential equation that describes the variation of temperature of the ball with time, T(/). Assume constant thermal conductivity and no heat generation in the object. Do not solve. 

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