Wall functions, temperature

In the previous section we discussed wall functions, which are used to reduce the number of cells. However, we must be aware that this is an approximation that, if the flow near the boundary is important, can be rather crude. In many internal flows—where all boundaries are either walls, symmetry planes, inlets, or outlets—the boundary layer may not be that important, as the flow field is often pressure determined. However, when we are predicting heat transfer, it is generally not a good idea to use wall functions, because the convective heat transfer at the walls may be inaccurately predicted. The reason is that convective heat transfer is extremely sensitive to the near-wall flow and temperature field. 

The heat flux to the wall and the wall temperature are related through a wall function... 

The wall temperature maps shown in Fig. 28 are intended to show the qualitative trends and patterns of wall temperature when conduction is or is not included in the tube wall. The temperatures on the tube wall could be calculated using the wall functions, since the wall heat flux was specified as a boundary condition and the accuracy of the values obtained will depend on their validity, which is related to the y+ values for the various solid surfaces. For the range of conditions in these simulations, we get y+ x 13-14. This is somewhat low for the k- model. The values of Tw are in line with industrially observed temperatures, but should not be taken as precise. 

An analysis and a physical interpretation of these observations will now be proposed. In order to analyse and classify the different heat transfer coefficient behaviours observed on figures 4 and 5 it is useful to represent, for a given vapour quality, the heat transfer coefficient as a function of the heat flux and the heat flux as a function of the wall-fluid temperature difference. For Dj, = 2 mm, since Co < 0.5, the results were analysed in terms of macroscale boiling. This was done on figure 8, which exhibits two trends ... 

The influence of a wall on the turbulent transport of scalar (species or enthalpy) at the wall can also be modeled using the wall function approach, similar to that described earlier for modeling momentum transport at the wall. It must be noted that the thermal or mass transfer boundary layer will, in general, be of different thickness than the momentum boundary layer and may change from fluid to fluid. For example, the thermal boundary layer of a high Prandtl number fluid (e.g. oil) is much less than its momentum boundary layer. The wall functions for the enthalpy equations in the form of temperature T can be written as ... 

As an example, consider a sphere of radius R with a black wall of temperature 4 flUed with an isothermal gas at 4, as shown in Figure 7.16. Neglect scattering and assume that the gas has a refractive index of 1 and an absorption coefficient k that is independent of the wavelength (gray assumption). The spectral intensity at the wall is a function of angle 0. From Equation (7.25), we have 4(0) = + 4 j(4)(1-c ). Since k is not a function of the wavelength,... 

There are many different types of thermocouples, depending on the temperature range needed. Table 5.1 shows the common thermocouple types and the alloys used in each. Each combination generates a specific millivolt output as a function of temperature. These millivolt reference tables are based on some reference junction temperature which is usually 0°C. Most electronic data acquisition systems have these tables built into the software, usually through a curve fit of the data, so the millivolt readings are automatically converted to temperature. For best accuracy, a thermocouple with the appropriate range should be used for the expected temperatures. For example, type T thermocouples are typically used for lower temperature measurements such as the water temperature in a cooling system. In industrial combustion. Type J and K thermocouples are commonly used to measure intermediate temperatures, such as the temperature inside a refractory wall or temperatures inside a burner. Type... 

Figure 10.28 Oxidative stability of samples taken from an unexposed MDPE pipe as a function of distance from the inner wall. Oxidation temperature measurements were obtained at a 10°Cmin scan rate and OIT at I90°C. With permission from Society of Plastic Engineers (Karlsson, Smith and Gedde 1992).

The employed oxygen is compressible fluid, and the flow between inlet and outlet is called isentropic flow. The pressure at the inlet is 0.9 MPa. The temperature is 293 K. The pressure at the outlet is 0.13 MPa. The flow speed around the wall is calculated through standard wall functions with consideration of no wall slippage. The flow field is calculated by the PISO which belongs to coupled algorithm with using Second Order Upwind as array of difference and VOF model as two phase flow. 

In the present eonfiguration, air inlet boundaries are assumed to be Pressure Inlet while outflow boundaries are assumed Pressure Outlet . Pressure inlet boundary conditions were used to define the total pressure and other scalar quantities at flow inlets. Pressure outlet boundary conditions were used to define the static pressure at flow outlets. At the nozzle inlet, the air pressure was varied. At the nozzle outlet, the pressure was supposed to be the external pressure (one atmosphere). At the wall of the nozzle standard wall function boundary condition was applied. Although the high velocity of air stream was a heat source that will increase the temperature in the nozzle, the nozzle length was very short and the process oecurs in a very short time. For simplification, it was assumed that the process is adiabatic i.e. no heat transfer occurred through walls. The flow model used was viscous, compressible airflow [1, 6-10]. The following series of equations were used to solve a compressible turbulent flow for airflow simulation [1,6-12] ... 

In the rotational core, used in HRE-1, the flow pattern tends to produce isotherms which are vertical cylinders. These are perturbed by boundary-layer mixing at the sphere walls. The temperature generally increases in the direction of the central axis, whieh is at outlet temperature. The gas bubbles are centrifuged rapidly into a gas void which forms at the center axis and from wliich gas can be removed. The gas void is quite stable in cores up to about 2 ft in diameter, l)ut in larger. spheres the pumping requirements to stabilize the void arc excessive [5]. The pressure drop through a rotational core is a function of the particular sj stem, but is usually above 5 inlet-velocity heads. 

Measured and calculated temperature profiles under melting lake ice, illustrates that large amounts of heat can be stored in the water beneath the ice before break-up. Fig. 2. This indicates that the molecular sub-layers below the ice interface are important, a fact that implies that wall functions need to be introduced in the turbulence modelling. 

It is still necessary to consider the role of entropy m irreversible changes. To do this we return to the system considered earlier in section A2.1.4.2. the one composed of two subsystems in themial contact, each coupled with the outside tliroiigh movable adiabatic walls. Earlier this system was described as a function of tliree independent variables, F , and 0 (or 7). Now, instead of the temperature, the entropy S = +. S P will be... 

The grand canonical ensemble is a set of systems each with the same volume V, the same temperature T and the same chemical potential p (or if there is more than one substance present, the same set of p. s). This corresponds to a set of systems separated by diathennic and penneable walls and allowed to equilibrate. In classical thennodynamics, the appropriate fimction for fixed p, V, and Tis the productpV(see equation (A2.1.3 7)1 and statistical mechanics relates pV directly to the grand canonical partition function... 

Electrodes or Langmuir probes may be inserted into plasmas that are large enough (>1 cm) and relatively cool (<10 K). The net current to the probe is measured as a function of the appHed voltage. Electron temperatures, electron and ion densities, and space and wall potentials may be derived from the probe signals. Interaction of plasmas with soHd probes tends to perturb plasma conditions. 

The Ubbelohde viscometer is shown in Figure 24c. It is particularly useful for measurements at several different concentrations, as flow times are not a function of volume, and therefore dilutions can be made in the viscometer. Modifications include the Caimon-Ubbelohde, semimicro, and dilution viscometers. The Ubbelohde viscometer is also called a suspended-level viscometer because the Hquid emerging from the lower end of the capillary flows down only the walls of the reservoir directly below it. Therefore, the lower Hquid level always coincides with the lower end of the capillary, and the volume initially added to the instmment need not be precisely measured. This also eliminates the temperature correction for glass expansion necessary for Cannon-Fen ske viscometers. 

For turbulent flow of a fluid past a solid, it has long been known that, in the immediate neighborhood of the surface, there exists a relatively quiet zone of fluid, commonly called the Him. As one approaches the wall from the body of the flowing fluid, the flow tends to become less turbulent and develops into laminar flow immediately adjacent to the wall. The film consists of that portion of the flow which is essentially in laminar motion (the laminar sublayer) and through which heat is transferred by molecular conduction. The resistance of the laminar layer to heat flow will vaiy according to its thickness and can range from 95 percent of the total resistance for some fluids to about I percent for other fluids (liquid metals). The turbulent core and the buffer layer between the laminar sublayer and turbulent core each offer a resistance to beat transfer which is a function of the turbulence and the thermal properties of the flowing fluid. The relative temperature difference across each of the layers is dependent upon their resistance to heat flow. 

The dimensionless relations are usually indicated in either of two forms, each yielding identical resiilts. The preferred form is that suggested by Colburn ran.s. Am. In.st. Chem. Eng., 29, 174—210 (1933)]. It relates, primarily, three dimensionless groups the Stanton number h/cQ, the Prandtl number c Jk, and the Reynolds number DG/[L. For more accurate correlation of data (at Reynolds number <10,000), two additional dimensionless groups are used ratio of length to diameter L/D and ratio of viscosity at wall (or surface) temperature to viscosity at bulk temperature. Colburn showed that the product of the Stanton number and the two-thirds power of the Prandtl number (and, in addition, power functions of L/D and for Reynolds number <10,000) is approximately equal to half of the Fanning friction fac tor//2. This produc t is called the Colburn j factor. Since the Colburn type of equation relates heat transfer and fluid friction, it has greater utility than other expressions for the heat-transfer coefficient. 

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