Temperature adiabatic 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... 

Since the kinetic energy is related to Ma, the adiabatic wall temperature might be reduced by a function of Ma for the cases where the viscous heat dissipation is negligibly small. Then, the values of Tw/Tstg for all channels are plotted as a function of Ma in Fig. 4.20. 

Variation of dimensionless adiabatic wall temperature with Prandtl... 

It is conventional to write this equation for the adiabatic wall temperature as ... 

Therefore, the adiabatic wall temperature is 36.9°C. Hence, since the wall is kepi at a temperature of 30°C, i.e., since Tw - TWgd is negative, heat is being transferred from the air to the plate. 

When viscous dissipation effects are important, however, there are three temperatures that have an influence on the temperature distribution and therefore on the fluid properties, these three temperatures being the wall temperature, the freestream temperature, and the adiabatic wall temperature, i.e., Tw, T, and TWmg. This is illustrated in Fig. 3.28. 

At this Reynolds number, the flow in the boundary layer will indeed be tutbulent so the assumed recovery factor is, in fact, the correct value. Hence, the adiabatic wall temperature is 159°C. 

At an altitude of 30,000 m the atmospheric pressure is approximately 1200 Pa and the temperature is approximately -4S°C. Assuming a turbulent boundary layer flow over an adiabatic flat plate, plot the variation of the adiabatic wall temperature with Mach number for Mach numbers between 0 and 5. 

Notice that the difference between the adiabatic wall temperature and the actual wall temperature is used in the definition so that the expression will yield a value of zero heat flow when the wall is at the adiabatic wall temperature. For gases with Prandtl numbers near unity the following relations for the recovery factor have been derived ... 

These recovery factors may be used in conjunction with Eq. (5-119) to obtain the adiabatic wall temperature. 

When very high flow velocities are encountered, the adiabatic wall temperature may become so high that dissociation of the gas will take place and there will be a very wide variation of the properties in the boundary layer. Eckert [4] recommends that these problems be treated on the basis of a heat-transfer coefficient defined in terms of enthalpy difference ... 

We must consider the laminar and turbulent portions of the boundary layer separately because the recovery factors, and hence the adiabatic wall temperatures, used to establish the heat flow will be different for each flow regime. It turns out that the difference is rather small in this problem, but we shall follow a procedure which would be used if the difference were appreciable, so that the general method of solution may be indicated. The free-stream acoustic velocity is calculated from... 

Air at 7 kPa and -40°C flows over a flat plate at Mach 4. The plate temperature is 35°C, and the plate length is 60 cm. Calculate the adiabatic wall temperature for the laminar portion of the boundary layer. 

A wind tunnel is to be constructed to produce flow conditions of Mach 2.8 at Ix = -40°C and p = 0.05 atm. What is the stagnation temperature for these conditions What would be the adiabatic wall temperature for the laminar and turbulent portions of a boundary layer on a flat plate If a flat plate were installed... 

We first calculate the recovery (adiabatic-wall) temperature. From Fig. 12-10... 

Figure 9.42 Radiation-affected adiabatic wall temperature.

Here Taw> commonly called the adiabatic wall temperature or the recovery temperature, is the equilibrium temperature the surface would attain in the absence of any heat transfer to or from the surface and in the absence of radiation exchange between the surroundings and the surface. In general the adiabatic wall temperature is dependent on the fluid properties and the properties of the bounding wall. Generally, the adiabatic wall temperature is reported in terms of a dimensionless recovery factor r defined as... 

The adiabatic wall temperature (recovery temperature) is given by... 

The recommended design method for Mach numbers up to supersonic (Ma < 4) and adiabatic wall conditions (Tw = Taw) is arbitrary since all the methods yield essentially the same results The design methods for hypersonic Mach numbers (Ma, > 4) and cold wall conditions (Tw< T w) should be based on conservatism. Comparison of the available skin friction data reveals differences of 20 percent on surfaces near adiabatic wall temperature and as much as a factor of 2 for highly cooled walls (T 0.2Taw). If only the most recent skin friction data for... 

The compressible-flow nature of RESS also affects the solution of Eq. (38). In particular, the wall temperature can be considerably higher than the bulk fluid temperature because as the high-speed fluid is brought to rest, the kinetic energy of the fluid is converted into internal energy. It should be emphasized that this temperature increase occurs even for a perfectly insulated channel wall with no external heating. For the compressible-flow case, Eckert (25) has shown that Eq. (38) can generally be used with the same heat transfer relations used for incompressible flow if the bulk fluid temperature is replaced with the adiabatic wall temperature Tad,w, so that... 

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