Steady heat conduction

FIG. 5-1 Temperature gradients for steady heat conduction in series through three solids. 

As shown in Fig. 21, in this case, the entire system is composed of an open vessel with a flat bottom, containing a thin layer of liquid. Steady heat conduction from the flat bottom to the upper hquid/air interface is maintained by heating the bottom constantly. Then as the temperature of the heat plate is increased, after the critical temperature is passed, the liquid suddenly starts to move to form steady convection cells. Therefore in this case, the critical temperature is assumed to be a bifurcation point. The important point is the existence of the standard state defined by the nonzero heat flux without any fluctuations. Below the critical temperature, even though some disturbances cause the liquid to fluctuate, the fluctuations receive only small energy from the heat flux, so that they cannot develop, and continuously decay to zero. Above the critical temperature, on the other hand, the energy received by the fluctuations increases steeply, so that they grow with time this is the origin of the convection cell. From this example, it can be said that the pattern formation requires both a certain nonzero flux and complementary fluctuations of physical quantities. 

The last term on the right-hand side can be obtained by solving the temperature profile in the solid bed. Consider a small, x-direction portion of the film and solid [Fig. 5.12(b)], We assume the solid occupies the region y > 5 (where 3 is the local film thickness) and moves into the interface with constant velocity vs.v. The problem thus reduces to a onedimensional steady heat-conduction problem with convection. In the solid, a steady, exponentially dropping temperature profile develops. The problem is similar to that in Section 5.4. The equation of energy reduces to... 

One-Dimensional Steady Heat Conduction 292 Boundary Conditions 294... 

Consider steady heat conduction through a large plane wall of thickness Alv = L and area A, as shown in Fig. 1 22. The temperature difference across the wall is AT =7 — 7V Experiments have shown that the rale of heat transfer Q through the wall is doubled when the temperature difference AT" across the wall or the area A normal to the direction of heat transfer is doubled, but is halved when the wall lliickness L is doubled. Thus we conclude that the rate of heat conduction through a plane layer is proportional to the temperature difference across the layer and the heal transfer area, but is inversely proportional to the thickness of the layer. That is. 

Transient and steady heat conduction in a plane wall. 

Note that we replaced the partial derivatives by ordinary derivatives in the one-dimensional steady heat conduction case since the partial and ordinary derivatives of a function are identical when the function depends on a single variable ordy [T = T x) in this case]. 

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

Starting with an energy balance on a ring-shaped volume element, derive the two-dimensional steady heat conduction equation in cylindrical coordinates for T(r, z) for the caf.se of constant thermal conductivity and no heat generation./... 

S4C Consider a. solid cylindrical rod whose side surface is maintained at a constant temperature while the end surfaces are perfectly insulated. The thermal conductivity of the rod material is constant and there is no heat generation. It is claimed that the temperature in the radial direction wiihin the rod will not vaty during steady heat conduction. Do you agree with this claim Why ... 

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. 

Consider steady heat conduction through the walls of a house during a winter day. We know that heat is continuously lost to the outdoors through the wall. Wc intuitively feel that heat transfer through the wall i.s in the normal direction to the wall surface, and no significant heat transfer takes place in the wall in other directions CFig. 3-1). 

Consider a plane wall of thickness L and average thermal conductivity k. The Isvo surfaces of the wall are maintained at constant temperatures of r, and T2. For one-dimensional steady heat conduction through the wall, we have 7(.v). Then Fourier s law of heat conduction for the wall can be expressed as... 

EXAMPLE5-1 Steady Heat Conduction in a Large Uranium Plate... 

Consider steady heat conduction in a plane wall whose left. surface (node 0) is maintained at 30°C while the right surface (node 8) is subjected to a heat flux of 1200 W/mT Express the finite difference formulation of the boundary nodes 0 and 8 for the case of no heal generation. Also obtain the finite dif-... 

C How does the finite difference formulation of a transient heat conduction problem differ from that of a steady heat conduction problem Wlial does Ihe term pAil.xCp(T — Tlij/Ar represent in the transient finite difference formulation ... 

In steady heat conduction, we generally have dqj/dxj = 0 as the plate is thin in the -direction, then 8 jdx2 = 0 and the differential equation for steady heat transfer looks... 

Consider steady heat transfer across a slab of material that extends infinitely in two directions and has thickness L. In the thin direction, x, the equation for steady heat conduction is... 

On the right hand side of equation (8.8) the second derivative of P appears and one obtains an equation similar to that describing steady heat conduction in the variables 0, H. There is also an analogy to the coefficient of thermal conductivity whose magnitude is determined by the velocity field. 

Sphere-Flat Test Results. Kitscha [47] performed experiments on steady heat conduction through 25.4- and 50.8-mm sphere-flat contacts in an air and argon environment at pressures between 10 5 torr and atmospheric pressure. He obtained vacuum data for the 25.4-mm-diameter smooth sphere in contact with a polished flat having a surface roughness of approximately 0.13 pm RMS. The mechanical load ranged from 16 to 46 N. The mean contact temperature ranged between 321 and 316 K. The harmonic mean thermal conductivity of the sphere-flat contact was found to be 51.5 W/mK. The emissivities of the sphere and flat were estimated to be e, = 0.2 and e2 = 0.8, respectively. 

To solve the nonlinear control equation (1) under the condition (2) approximately using FEM, we need to establish relevant functional. The paper adopts one-dimensional nonlinear FEM to solve the above-mentioned one-dimensional heat conduction problem. Under the condition of assumptions in this paper, the element functional [8] (5.14) of one-dimensional steady heat conduction problem under the convective heat transfer boundary condition is... 

We now consider bar element, and the element length is f. Two nodes are denoted by i,j. The trial function of temperature field is linear distribution. Under the convective heat transfer boundary condition, the finite element basic equation of steady heat conduction in the three-layered composite plate is [8]... 

---