Steady-state heat conduction

The thermal conductivity of a material is defined in terms of the transport of heat under steady-state conditions. On the other hand, one is often interested in the transport of heat when a specimen is not at equilibrium so that the flow of heat is transient. The thermal diffusivity a , which is defined by Equation 14.2, describes these time-dependent, non-steady-state aspects of heat flow. The thermal diffusivity is used to calculate the temperature (T) as a function of the position within the specimen (z) and the time (t) under non-steady-state conditions. It is related by Equation 14.3 to the thermal conductivity, the density, and the specific heat capacity. The values of X and a can be measured independently. However, often one of them (usually a) is estimated from the measured value of the other one (usually X) by using Equation 14.3. If X is in J/(K m sec), cp is in J/(g K) and p is in g/cc, then the a value calculated by using Equation 14.3 must be multiplied by 100 to convert it into our preferred diffusion units of cm2/sec. 

Equation (11.1) is essentially a solution of Eq. (11.7) and is based on a few assumptions and simplifications, e.g., no axial heat conduction, constant average heat conductivity and specific heat, constant heat source, steady-state heat transfer, one-dimensional (radial) heat flux, cylindrical geometry in the waste and in the surrounding material, e.g., salt, and no heat source in the salt. 

Demonstration of the creation of multilayer line structures in the seal will result in the collection of physical measurement data on the change in seal features during heat up, steady state operation, transients, or on cool down. A radar type image of the signal characteristics will show build up or relaxation of stresses, discontinuities in seal structure including porosity, delamination of the seal from either the cell or interconnect, and ultimately the catastrophic failure of the seal. Additionally with the ability to use a metal interconnect as one of the plane references coupled with a trace on the oxide coating on the interconnect, electrical conductivity and structural integrity of the thin oxide film on the metal interconnect can be determined within the vicinity of the trace. 

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. 

Steady state pi oblems. In such problems the configuration of the system is to be determined. This solution does not change with time but continues indefinitely in the same pattern, hence the name steady state. Typical chemical engineering examples include steady temperature distributions in heat conduction, equilibrium in chemical reactions, and steady diffusion problems. 

The prototype elhptic problem is steady-state heat conduction or diffusion,... 

Example 1. Steady-State Conduction with Heat Generation. 5-10... 

In the simplest case of one-dimensional steady flow in the x direction, there is a parallel between Eourier s law for heat flowrate and Ohm s law for charge flowrate (i.e., electrical current). Eor three-dimensional steady-state, potential and temperature distributions are both governed by Laplace s equation. The right-hand terms in Poisson s equation are (.Qy/e) = (volumetric charge density/permittivity) and (Qp // ) = (volumetric heat generation rate/thermal conductivity). The respective units of these terms are (V m ) and (K m ). Representations of isopotential and isothermal surfaces are known respectively as potential or temperature fields. Lines of constant potential gradient ( electric field lines ) normal to isopotential surfaces are similar to lines of constant temperature gradient ( lines of flow ) normal to... 

The overall heat transfer coefficient, U, is a measure of the conductivity of all the materials between the hot and cold streams. For steady state heat transfer through the convective film on the outside of the exchanger pipe, across the pipe wall and through the convective film on the inside of the convective pipe, the overall heat transfer coefficient may be stated as ... 

Liquified gases are sometimes stored in well-insulated spherical containers that are vented to the atmosphere. Examples in the industry are the storage of liquid oxygen and liquid ammonia in spheres. If the radii of the inner and outer walls are r, and r, and the temperatures at these sections are T, and T, an expression for the steady-state heat loss from the walls of the container may be obtained. A key assumption is that the thermal conductivity of the insulation varies linearly with the temperature according to the relation ... 

In steady-state conditions the right side of Eq. (4.180) is zero, and no heat generation takes place the thermal conductivity in the one-dimensional case is constant. The solution of Eq. (4.182) is... 

I FIGURE 11.27 Heat conduction through an external wall. The temperature distribution over die wall thickness is linear only under steady-state conditions. 

Under steady-state conditions, the temperature distribution in the wall is only spatial and not time dependent. This is the case, e.g., if the boundary conditions on both sides of the wall are kept constant over a longer time period. The time to achieve such a steady-state condition is dependent on the thickness, conductivity, and specific heat of the material. If this time is much shorter than the change in time of the boundary conditions on the wall surface, then this is termed a quasi-steady-state condition. On the contrary, if this time is longer, the temperature distribution and the heat fluxes in the wall are not constant in time, and therefore the dynamic heat transfer must be analyzed (Fig. 11.32). 

In reality, heat is conducted in all three spatial dimensions. While specific building simulation codes can model the transient and steady-state two-dimensional temperature distribution in building structures using finite-difference or finite-elements methods, conduction is normally modeled one-... 

Conduction of heat through plain surfaces under steady-state conditions is given by the product of the area, temperature difference, and overall conductance of the surface (see Section 1.8) ... 

For certain products, skill is required to estimate a product s performance under steady-state heat-flow conditions, especially those made of RPs (Fig. 7-19). The method and repeatability of the processing technique can have a significant effect. In general, thermal conductivity is low for plastics and the plastic s structure does not alter its value significantly. To increase it the usual approach is to add metallic fillers, glass fibers, or electrically insulating fillers such as alumina. Foaming can be used to decrease thermal conductivity. 

Temperature gradients within the porous catalyst could not be very large, due to the low concentration of combustibles in the exhaust gas. Assuming a concentration of 5% CO, a diffusion coefficient in the porous structure of 0.01 cms/sec, and a thermal conductivity of 4 X 10-4 caI/sec°C cm, one can calculate a Prater temperature of 1.0°C—the maximum possible temperature gradient in the porous structure (107). The simultaneous heat and mass diffusion is not likely to lead to multiple steady states and instability, since the value of the 0 parameter in the Weisz and Hicks theory would be much less than 0.02 (108). 

Mass transfer from a single spherical drop to still air is controlled by molecular diffusion and. at low concentrations when bulk flow is negligible, the problem is analogous to that of heat transfer by conduction from a sphere, which is considered in Chapter 9, Section 9.3.4. Thus, for steady-state radial diffusion into a large expanse of stationary fluid in which the partial pressure falls off to zero over an infinite distance, the equation for mass transfer will take the same form as that for heat transfer (equation 9.26) ... 

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. 

Example 5.8 Suppose that, to achieve a desired molecular weight, the styrene polymerization must be conducted at 413 K. Use external heat transfer to achieve this temperature as the single steady state in a stirred tank. 

The numerical experiment started at a steady-state value of 200 C for both temperature nodes with an output of 16.89% for both heaters output number 1 was then stepped to 19.00%. If both outputs had been stepped to 19%, then both nodes would have gone to 220 C. The temperature of node 5 does not go as high, and the temperature of node 55 goes too high. In the reduced order model, the time constant x represents the effect of radial heat conduction, while the time constant X2 represents the effect of axial heat conduction. SimuSolv estimates these two parameters of the dynamic model as ... 

In the articles cited above, the studies were restricted to steady-state flows, and steady-state solutions could be determined for the range of Reynolds numbers considered. Experimental work on flow and heat transfer in sinusoidally curved channels was conducted by Rush et al. [121]. Their results indicate heat-transfer enhancement and do not show evidence of a Nusselt number reduction in any range... 

Here, q is the flux of heat (W m ), X is the thermal conductivity (W K ), T is temperature (K), and r is the distance from the center of the spherical heat source. Under the steady state approximation, the heat generated in the small sphere, Qj , is equal to the heat flow, Qflow. from the surface of the small sphere to the surrounding medium, as expressed by Eq. (8.7). 

The following example, taken from Welty et al. ( 1976), illustrates the solution approach to a steady-state, one-dimensional, diffusional or heat conduction problem. 

A metal rod is in contact with a constant temperature source at each end. At steady state the heat conducted towards the center is balanced by the heat loss by radiation. This leads to a symmetrical temperature profile in the rod, as shown. 

This result for the most likely change in moment is equivalent to Fourier s law of heat conduction. To see this take note of the fact that in the steady state the total rate of change of moment is zero, E = 0, so that the internal change is... 

The generalized form for steady-state heat conduction across a thin film (where we allow the insulating material a thickness 8) is given by... 

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