Conductive Heat Transfer—Steady State

Conduction is a process where heat flows within a body (soUd, liquid, or gas) from a region of high temperature to one of lower temperature. While the basic mechanism of heat conduction is different for metals, non-metals, and fluids, the result is the same—an increase in the vibrational amplitude as heat flows into a body. The basic law of conductive heat transfer is Fourier s equation (1822)  

In general, values of Ic vary with the temperature, but in most problems, this variation may be neglected by using a value corresponding to the mean temperature. The Btu is a unit of thermal energy, and in terms of the fundamental set of dimensions (F, L,T, 6) it may be expressed as [FL. Therefore, the dimensions of Jc become  

Representative values of k for different materials are given in Table 11.4 together with other thermal properties. 

For a steady state (no change with time), and assuming a constant value of k with temperature, Eq. (11.22) may be written as follows for a plane slab where the temperature of the hot and cold walls are T2 and Fj respectively, and b is the slab thickness. 

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For the analysis, a steady-state fire was assumed. A series of equations was thus used to calculate various temperatures and/or heat release rates per unit surface, based on assigned input values. This series of equations involves four convective heat transfer and two conductive heat transfer processes. These are ... 

Usually, the rate of heat transfer is a combination of conduction and convection in a heat exchanger system as illustrated in Fig. 7.1 and only the fluid temperature on either side of the solid surface is known. For steady state, the rate of conduction heat transfer and the rate of convection heat transfer are equal. The total resistance (R) of the combined rate of heat transfer is... 

There is a myriad of analytical solutions for steady-state conduction heat-transfer problems available in the literature. In this day of computers most of these solutions are of small utility, despite their exercise in mathematical facilities. This is not to say that we cannot use the results of past experience to anticipate answers to new problems. But, most of the time, the problem a person wants to solve can be attacked directly by numerical techniques, except when there is an easier way to do the job. As a summary, the following suggestions are offered ... 

Figure 3.47 shows the evolution of the heating process of the composite block and how it attains a complex steady state structure with the surface zones covered by complicated isothermal curves (see also Fig. 3.46). Secondly, this figure shows how the brick with the higher thermal conductivity is at steady state and remains the hottest during the dynamic evolution. As explained above, this fact is also shown in Fig. 3.46 where all high isothermal curves are placed in the area of the brick with highest thermal conductivity. At the same time an interesting vicinity effect appears because we observe that the brick with the smallest conductivity does not present the lowest temperature in the centre (case of curve G compared with curves A and B). The comparison of curves A and B, where we have X = 0.2, with curves C and D, where X = 0.4, also sustains the observation of the existence of a vicinity effect. In Fig. 3.48, we can also observe the effect of the highest thermal conductivity of one block but not the vicinity effect previously revealed by Figs. 3.46 and 3.47. If we compare the curves of Fig. 3.47 with the curves of Fig. 3.48 we can appreciate that a rapid process evolution takes place between T = 0 and T = 1. Indeed, the heat transfer process starts very quickly but its evolution from a dynamic process to steady state is relatively slow.

The temperature field in a tissue is determined by heat conduction and convection, metabolic heat generation, thermal energy transferred to the tissue from an external source or the surrounding tissue, and the tissue geometry. Thermal conduction is characterized by a thermal conductivity, k, at steady state and by a thermal diffusivity, a, in transient state. Thermal convection is characterized by the topology of the vascular bed and the blood flow rate, which is subject to the thermal regulation. 

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. 

At time th the heat flow by conduction matches the steady-state convective heat transfer from the body. If conditions are met to initiate convection before t, (this will depend on Pr and Ra), the heat flow falls monotonically in the transition regime, as along path A in Fig. 4.37b. Otherwise convection will not be initiated until to > t so the heat flow will have fallen below the steady-state value and must therefore recover from the undershoot in the transition regime as shown by path B. 

In the transition regime (t0 < x < t ), if the conduction heat transfer at xD exceeds the steady-state heat transfer for high Ra, Nu falls monotonically as shown in Fig. 4.39. For small... 

Conduction heat transfer mechanisms in flame impingement (a) steady-state, and (b) transient. 

From a heat balance at the vapor-liquid interface, it is found that the net mass-transfer energy must equal the difference between the heat transferred from the gas to the interface and the heat transferred from the interface through the liquid. This difference is represented by the previously developed natural convection heat-transfer relation for the gas-phase portion of the balance (7) and by a moving boundary steady-state conduction heat-transfer relation for the liquid-phase portion of the balance. This leads to the expression... 

A steady decrease in cryosurface pumping speed during the latter part of each run was attributed to an increase in the cryosurface temperature due to cryodeposit build-up. Data were correlated by a steady-state conduction heat transfer analysis which also gave good correlation between the hot and cold CO2 data. 

The use of a numerical heat transfer model and a design optimization procedure to simulate and synthesize the heater configuration in a laboratory-scale pultrusion die was developed and studied by Awa and West (1992). A two-dimensional steady-state conduction heat transfer model was developed to compute the temperature profile within the laboratory-scale die. 

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 ... 

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). 

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) ... 

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. 

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... 

We follow the analysis of Frank-Kamenetskii [3] of a slab of half-thickness, rG, heated by convection with a constant convective heat transfer coefficient, h, from an ambient of Too. The initial temperature is 7j < 7 ,XJ however, we consider no solution over time. We only examine the steady state solution, and look for conditions where it is not valid. If we return to the analysis for autoignition, under a uniform temperature state (see the Semenov model in Section 4.3) we saw that a critical state exists that was just on the fringe of valid steady solutions. Physically, this means that as the self-heating proceeds, there is a state of relatively low temperature where a steady condition is sustained. This is like the warm bag of mulch where the interior is a slightly higher temperature than the ambient. The exothermiscity is exactly balanced by the heat conducted away from the interior. However, under some critical condition of size (rG) or ambient heating (h and Too), we might leave the content world of steady state and a dynamic condition will... 

In the common case of cylindrical vessels with radial symmetry, the coordinates are the radius of the vessel and the axial position. Major pertinent physical properties are thermal conductivity and mass diffusivity or dispersivity. Certain approximations for simplifying the PDEs may be justifiable. When the steady state is of primary interest, time is ruled out. In the axial direction, transfer by conduction and diffusion may be negligible in comparison with that by bulk flow. In tubes of only a few centimeters in diameter, radial variations may be small. Such a reactor may consist of an assembly of tubes surrounded by a heat transfer fluid in a shell. Conditions then will change only axially (and with time if unsteady). The dispersion model of Section P5.8 is of this type. 

Unlike Fourier s Law, Eq. (4.62) is purely empirical—it is simply the definition for the heat transfer coefficient. Note that the units of he (W/m -K) are different from those for thermal conductivity. Under steady-state conditions and assuming that the heat transfer area is constant and h is not a function of temperature, the following form of Eq. (4.62) is often employed ... 

Figure 5.2 shows the temperature gradients in the case of heat transfer from fluid 1 to fluid 2 through a flat metal wall. As the thermal conductivities of metals are greater than those of fluids, the temperature gradient across the metal wall is less steep than those in the fluid laminar sublayers, through which heat must be transferred also by conduction. Under steady-state conditions, the heat flux q (kcal In m 2 or W m ) through the two laminar sublayers and the metal wall should be equal. Thus,... 

Heat Transfer by Conduction. In the theoretical analysis of steady state, heterogeneous combustion as encountered in the burning of a liquid droplet, a spherically symmetric model is employed with a finite cold boundary as a flame holder corresponding to the liquid vapor interface. To permit a detailed analysis of the combustion process the following assumptions are made in the definition of the mathematical model ... 

Thermal conductivity may be defined as the quantity of heat passing per unit time normally through unit area of a material of unit thickness for unit temperature difference between the faces. In the steady state, i.e. when the temperature at any point in the material is constant with time, conductivity is the parameter which controls heat transfer. It is then related to the heat flow and temperature gradient by ... 

Very little data is available on the measurement of heat transfer coefficient. Hands15 mentions the empirical nature of the coefficient and the numerous factors which will affect its value, particularly between rubber and a fluid. Griffiths and Norman66 calculated the heat transfer coefficients for rubbers in air and water. Hall et al67 investigated the effect of contact resistance on steady state measurements of conductivity. 

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