Isothermal wall

Two-dimensional compressible momentum and energy equations were solved by Asako and Toriyama (2005) to obtain the heat transfer characteristics of gaseous flows in parallel-plate micro-channels. The problem is modeled as a parallel-plate channel, as shown in Fig. 4.19, with a chamber at the stagnation temperature Tstg and the stagnation pressure T stg attached to its upstream section. The flow is assumed to be steady, two-dimensional, and laminar. The fluid is assumed to be an ideal gas. The computations were performed to obtain the adiabatic wall temperature and also to obtain the total temperature of channels with the isothermal walls. The governing equations can be expressed as... 

Dynamics of a Vaporizing a (10-50 (mu) Water) Droplet in Laminar Entry Region of a Straight Channel (with Isothermal Walls, as in a Dry Cooling Tower)... 

C per 10 MPa. The temperature rise in laminar flow is highly nonuniform, being concentrated near the pipe wall where most ofthe dissipation occurs. This may result in significant viscosity reduction near the wall, and greatly increased flow or reduced pressure drop, and a flattened velocity profile. Compensation should generally be made for the heat effect when AP exceeds 1.4 MPa (203 psi) for adiabatic walls or 3.5 MPa (508 psi) for isothermal walls (Gerard, Steidler, and Appeldoorn, Ind. Eng. Chem. Fundam., 4, 332-339 [1969]). 

Galili and Takserman-Krozer (20) have proposed a simple criterion that signifies when nonisothermal effects must be taken into account. The criterion is based on a perturbation solution of the coupled heat transfer and pressure flow isothermal wall problem of an incompressible Newtonian fluid. 

The system of Eqs. 13.3-17 and 13.3-18 can be solved for the adiabatic, isothermal, or constant wall flux cases using the Crank-Nicolson method. The thermomechanical and reaction data for such systems were evaluated by Lifsitz, Macosko, and Mussatti (99) at 45°C for a polyester triol and a chain extended 1,6-hexamethylene diisocyanate (HDI) with dibutyltin as a catalyst. Figure 13.46 gives the temperature profiles for the isothermal-wall case. Because of the high heat of polyurethane formation and the low conductivity of... 

Fig. 13.46 Temperature distribution in a reacting polyurethane slab isothermal-wall simulations. Dotted line denotes the adiabatic temperature rise and x s indicate gel points, k = 1, n — 1, AT — 0.423, and B = 18.7, 4>gel = 0.707. [Reprinted by permission from E. Broyer and C. W. Macosko, Heat Transfer and Curing in Polymer Reaction Molding, AICHE J. 22, 268 (1976).]...

The usual limitations (isothermal wall, radicals quasi-steady state, constant pressure) found in the lit -erature for similar models were released, and the importance of correctly evaluating the propagation and termination rate constants, kp and kt, was shown. 

Consider the natural convective flow of air at 10°C though a plane vertical channel with isothermal walls whose temperature is 40°C and whose height is 10 cm. Determine how the mean heat transfer rate from the heated walls varies with the gap between the walls. 

Fig. B-2 Temperature profiles in laminar boundary layer with isothermal wall.

The low-dimensional model for non-isothermal wall-catalyzed reaction in a tubular reactor is given by Eqs. (283)-(285) and... 

Figure 3.62 shows the temperature field of a quarter of the radial section of the reactor before the reaction firing. Combining the values of Tq, T, and Bi results in an effective cooling of the reactor near the walls during the initial instants of the reaction (T = 0 — 0.05). In Fig. 3.63 is shown the temperature field when the dimensionless time ranges between T = 0.05 and T = 0.11. Here, the reaction runaway starts and we can observe that an important temperature enhancement occurs at the reactor centre, at the same time the reactant conversion increases (Fig. 3.64). The evolution of the reaction firing and propagation characterize this process as a very fast process. We can appreciate in real time that the reaction is completed in 10 s. It is true that the consideration of isothermal walls can be criticized but it is important to notice that the wall temperature is not a determining factor in the process evolution when the right input temperature and the right input concentrations of reactants have been selected.

We will now investigate radiative exchange between the isothermal walls (zones) of the enclosure illustrated in Fig. 5.57. The temperature of some of the zones is known, for others the heat flow supplied from or released to the outside is given. The heat flows of the zones with known temperatures and the temperature of each zone with stipulated heat flow are what we are seeking. There are as many unknown quantities (temperatures or heat flows) as there are zones. 

In complicated geometries the boundary walls of an enclosure must be divided into several zones. Non-isothermal walls also have to be split into a number of isothermal surfaces (= zones) in order to increase the accuracy of the results13. The equivalent electrical circuit diagram introduced in 5.5.3.2 would be confusing for this case. It is more sensible to set up and then solve a system of linear equation for the n radiosities of the n zones. The difficulty here is not the solving of the large number of equations in the system, but is the determination of the n2 view factors that appear. 

An isothermal gas at temperature TG is enclosed by isothermal walls at a temperature Tw < Tg. As a simplification the walls will be idealised as black bodies reflection does not need to be considered. A heat flow QGW from the gas to the colder walls is transferred by radiative exchange. If the gas is to maintain its... 

Let us consider the problem of dissipative heating of a fluid in a plane channel of width 2h with isothermal walls on which the same constant temperature is... 

Filmwise condensation (or evaporation) involves heat transfer to the liquid-vapor interface as well as convective flow in either or both phases. Here, we consider steady vapor condensation at the saturation temperature Ts which forms a liquid film while flowing down next to a vertical isothermal wall at Ta < Ts (Fig. 113). Assuming a continuous smooth film starting at x = 0, we wish to determine the variation in film thickness S(x) and the local Nusselt number. 

Figure 113 FSlmwise condensation of saturated vapor along a vertical isothermal wall.

Isothermal Wall. Natural convection also affects the laminar thermal development in a tube with an isothermal wall. In this case the temperature differences in the fluid near the tube inlet initiate a natural convection motion, but as the fluid temperature approaches the wall temperature far downstream, the motion slows and the fully developed Nusselt number (Nujr = 3.66) is approached. 

For some systems, heat input to different regions of the surface is adjusted to obtain a specific boundary condition—often to approximate an isothermal wall. This is most easily done with a number of small heating elements that can be individually adjusted to maintain a constant wall temperature. The local heat flux, or at least the heat flux averaged over the size of each individual heater, can be determined from the power input to the heaters. These heaters can be placed quite close to the surface. 

For an isothermal wall, when a simulated particle collides with the wall, a diffuse-reflection model is used to determine the result of reflection, whereby the outgoing velocity is randomly assigned according to a half-range Maxwellian distribution determined by the wall temperature. This is also known as the full thermal and momentum accommodation method. 

From a modeling perspective, then, a reasonable first approximation of heat transfer effects is the imposition of isothermal boundary conditions on the micro-nozzle walls. The particular wall temperature would reflect the thermal state of the spacecraft as a whole based oti its orbital location with respect to the Sun. To demonstrate the effect of heat transfer, we present findings obtained from numerical simulations of the NASA/GSFC linear micro-nozzle for expansion angles of 15° and 30° and isothermal wall temperatures ranging from 50 to 1,(XX) K. 

The thrust production (normalized with respect to quasi-1D inviscid theory) as a function of isothermal wall temperature is shown in Fig. 7. For comparison, the adiabatic results are also given. 

The observed effects of heat transfer on the flow in micro-nozzles are readily explained as follows. From compressible Rayleigh flow, it is known that removing heat from a supersonic flow acts to accelerate the flow. At steady-state, the bulk of the flow in the micro-nozzle expander is supersonic, and thus, heat transfer acts to further accelerate the supersonic flow. Concurrently, as the flow is cooled, the exit density p increases. The overall effect is an increase in thrust. Heat extraction from the flow into the substrate increases performance from the subsonic layer point of view as well. For low nozzle wall temperatures, the local sonic velocity is diminished and the near-wall Mach number increases. This phenomenon is the force driving the reduction in subsonic layer size for micro-nozzle flows with heat removal. In fact, with sufficient heat extraction from the flow, the subsonic layer can be reduced to the point where the competing effects of viscous forces and nozzle geometry cause the optimum expander half angle to be shifted from 30° to a more traditional expander half angle of 15°. This is demonstrated in Fig. 7 for isothermal wall temperatures less than 700 K. 

Supersonic Micronozzles, Fig. 7 Steady-state micro-nozzle performance including heat transfer through an isothermal wall. A reduced wall temperature is used to model heat losses from the flow field into the silicon micro-nozzle substrate acting as a thermal sink. As heat is removed from the expander, the supersonic flow is further accelerated, subsonic layer size is reduced, and performance as measured by thrust output increases (Note that the adiabatic results nearly coincide with the 900 K isothermal wall values as indicated)... 

The shape of the velocity profiles with one adiabatic wall is considerably different from the shape with two isothermal walls. In the latter case the velocity profile has a typical s-shape while in the former case there is monotonic reduction in the slope of the curves when A > 1. The velocities with an adiabatic screw are higher than with an isothermal screw. As a result, the flow rate is increased considerably compared to the isothermal drag flow rate. When A = 1 the flow rate equals 0.5 the flow rate increases with the value of A. When A becomes very large, the flow rate approaches unity and the velocity profile approaches plug flow. 

Because the die wall material usually has a thermal conductivity much higher than polymer melts, adiabatic conditions are not likely to be achieved. On the other hand, it is also not likely that the wall temperature will remain constant. In this case, the heat flux through the wall would be such as to maintain a perfectly constant temperature along the wall. This is referred to as an isothermal wall boundary condition. Because of the high thermal conductivity of the wall, the isothermal boundary condition is more likely to occur than the adiabatic boundary condition. Adiabatic conditions can be approached if the die is very well insulated. In most actual cases, the true thermal boundary condition will be somewhere between isothermal and adiabatic, depending on the design of the die and external conditions around the die. A typical temperature profile resulting from the velocity profiles shown in Fig. 7.106 is shown in Fig. 7.108. 

Equation 7.442 is a very useful relationship for determining fully developed temperature profiles in pipe flow law fluids. The maximum fully developed temperature always occurs at the center of the flow channel as can be seen from Eq. 7.442 as well as from Fig. 7.111, which illustrates the fully developed temperature profile under isothermal wall conditions and at various values of the power law index. 

Fully developed temperature profiles, isothermal wall conditions... 

Precise comparisons between packed beds and ceramic foam structures are complex since many factors - activity, effectiveness factors, mass and heat transfer, and pressure drop - are all interdependent. We have simulated the performance of a conventional steam reformer and compared it to one containing a ceramic foam cartridge loaded to achieve equivalent intrinsic activity per gram of catalyst. A 1-D model developed and tested previously for heat-pipe reformers with isothermal walls was used [57]. Pressure drop, mass transfer and heat transfer correlations for the packed bed were known to be accurate for commercial catalysts those used for the foam were determined in the studies described above. Process conditions and results are given in Table 7. 

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