Difference equations heat flux

Some materials exhibit nearly steady mass loss rates when exposed to a fixed radiant heat flux. The surface temperature for these materials reaches a steady value after a short initial transient period, and all terms in Equation 14.7 are approximately constant at a specified heat flux level. L can then be obtained by measuring steady mass loss rates at different radiant heat flux levels, and... 

The driving force of the heat transfer process is the tendency of systems to equilibrate out temperature differences. The heat flux = Q/A is directly proportional to the temperature difference AT = Tb — T , the proportionality factor being the heat transfer coefficient h, which is also defined by the general heat transfer equation ... 

Seldom is the temperature difference across the wall thickness of an item of equipment known. Siace large temperature gradients may occur ia the boundary layers adjacent to the metal surfaces, the temperature difference across the wall should not be estimated from the temperatures of the fluids on each side of the wall, but from the heat flux usiag equation 27... 

For large temperature differences different equations are necessary ana usually are specifically applicable to either gases or liquids. Gambill (Chem. Eng., Aug. 28, 1967, p. 147) provides a detailed review of high-flux heat transfers to gases. He recommends... 

An alternative metlrod of solution to these analytical procedures, which is particularly useful in computer-assisted calculations, is the finite-difference technique. The Fourier equation describes the accumulation of heat in a thin slice of the heated solid, between the values x and x + dx, resulting from the flow of heat tlirough the solid. The accumulation of heat in the layer is the difference between the flux of energy into the layer at x = x, J and the flux out of the layer at x = x + dx, Jx +Ox- Therefore the accumulation of heat in the layer may be written as... 

The coefficients in the equations differ slightly in different references, depending on the entrainment coefficients used. The convective heat flux , in W or W/m from the heat source, can be estimated from the energy consumption of the heat source by... 

In the finite-difference appntach, the partial differential equation for the conduction of heat in solids is replaced by a set of algebraic equations of temperature differences between discrete points in the slab. Actually, the wall is divided into a number of individual layers, and for each, the energy conserva-tk>n equation is applied. This leads to a set of linear equations, which are explicitly or implicitly solved. This approach allows the calculation of the time evolution of temperatures in the wall, surface temperatures, and heat fluxes. The temporal and spatial resolution can be selected individually, although the computation time increa.ses linearly for high resolutions. The method easily can be expanded to the two- and three-dimensional cases by dividing the wall into individual elements rather than layers. 

The family of uniformly heated round tubes has the simplest heat flux distribution and test-section geometry for which burn-out may occur, and it was selected for the preliminary investigation to test the different sets of important properties. Now, for a uniformly heated tube, the burn-out flux for a given fluid can be represented by the equation... 

Another important case is where the heat flux, as opposed to the temperature at the surface, is constant this may occur where the surface is electrically heated. Then, the temperature difference 9S — o will increase in the direction of flow (x-direction) as the value of the heat transfer coefficient decreases due to the thickening of the thermal boundary layer. The equation for the temperature profile in the boundary layer becomes ... 

According to Ivey (1967), the above equation may be in different forms in three different regions [Sec. 2.2.5.2, Eqs. (2-66), (2-66a), and (2-666)]. Since /in Eq. (2-97) is interpreted as an average /over the whole heating surface, and since heat flux is not a strong function of the frequency, a best single approximation applicable in the whole range of interest is used (Cole, 1967) ... 

Heat Transfer to the Containing Wall. Heat transfer between the container wall and the reactor contents enters into the design analysis as a boundary condition on the differential or difference equation describing energy conservation. If the heat flux through the reactor wall is designated as qw, the heat transfer coefficient at the wall is defined as... 

The basic hypothesis is that the temperature profiles of the two flows are parallel. This means that the heat exchanged at the hot side equals that exchanged at the cold side. This is an approximation since extracting electric power from the heat flux implies a difference between the two heat fluxes. However, the hypothesis, with only a minor influence on the final result, makes the equations compact and easy to manage. 

Thermal Conductivity of Laminar Composites. In the case of laminar composites or layered materials (cf. Figure 1.74), the thermal conductance can be modeled as heat flow through plane walls in a series, as shown in Figure 4.36. At steady state, the heat flux through each wall in the x direction must be the same, qx, resulting in a different temperature gradient across each wall. Equation (4.2) then becomes... 

Depending on how a solution is to be used or interpreted, there may be different requirements for its accuracy. Say an energy equation is solved to determine a temperature field. Evaluating the heat flux at a boundary depends on evaluating. the temperature gradient. When information about the solution s derivatives is important, the solution itself usually must be determined more accurately than if derivative information is not needed. 

In addition to simply solving the differential equation, we seek to use the solution to understand and quantify the heat transfer between the fluid and the duct walls. The heat flux q" (W/m2) can be described in terms of a heat-transfer coefficient h (W/m2 K), with the thermal driving potential being the difference between the wall temperature and the mean fluid temperature ... 

We find the state of the reaction products at x 0, the point from which we perform the integration, from the conservation laws. These same laws are contained in the differential equations. In the general case of differing D and k there are no simple relations between the concentrations and the temperature. However, the fluxes of various types of molecules and the heat flux are interrelated by equations (3.2) and (3.3). Therefore, if one of the conditions (3.6), (3.8) is satisfied, then the others will prove to be identically satisfied. Varying only the quantity u, we must succeed in satisfying one condition, which is always possible. 

For the reformer we assume that the outer wall temperature profile of the reformer tubes decouples the heat-transfer equations of the furnace from those for the reformer tubes themselves. The profile is correct when the heat flux from the furnace to the reformer tube walls equals the heat flux from the tube walls to the reacting mixture. We must carry out sequential approximating iterations to find the outer wall temperature profile Tt,o that converges to the specific conditions by using the difference of fluxes to obtain a new temperature profile T) o for the outer wall and the sequence of calculations is then repeated. In other words, a T) o profile is assumed to be known and the flux Q from the furnace is computed from the equations (7.136) and (7.137), giving rise to a new Tt o-This profile is compared with the old temperature profile. We iterate until the temperature profiles become stationary, i.e., until convergence. 

Comparing Equation (4) with Equation (5), i = qy, AV = AT and Rn = (bn/kn). Thus, we can think of the (b/k) values as thermal resistances, the resistance of the material layer to heat transfer by conduction. From Equation (4), it can be seen that the heat flux is equal to the overall temperature difference AT, divided by the sum of the thermal resistance of each layer L(b/k). The concept of thermal resistance of a layer is very useful and we will return to it later. 

The test procedure described in ASTM E 1678 minimizes the number of animal tests. In this test procedure, a specimen is exposed to a radiant heat flux of 50kW/m2, and the products of combustion are collected in a 0.2 m3 chamber. Test duration is 30 min. Additional tests are performed with specimens of different size to find the exposed area that is expected to result in 50% lethality of test animals exposed over the 30-min test duration to the atmosphere in the chamber. The lethality is determined on the basis of analytical measurements of the composition of the contents of the chamber and the N-Gas model (see Equation 14.20). To verify the results, two additional tests are conducted with 70% and 140% of that specimen area and six rats exposed to the gases in the chamber. 

Taking into account the different heat fluxes in the cone calorimeter setup L/ ir = < " + dfLp - ML - gL), Equation 4.1 was proposed for the idealized steady-state HRR (HRRst) during a steady-state burning 60 62 107108... 

In this section, the correlation in the heat flux ratio versus the pyrolyzed depth given by Equation 19.14 is incorporated into the numerical model to predict the pyrolysis process of the PA6 nanocomposite at different heat fluxes and thicknesses. The boundary conditions remain as those given by Equations 19.8 and 19.9. However, the MLR is now calculated from the heat flux ratio correlation in Equation 19.14 as... 

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