Cross sectional average temperature

The temperature of the fluid t F far away from the wall, appears in (1.23), the definition of the local heat transfer coefficient. If a fluid flows around a body, so called external flow, the temperature t F is taken to be that of the fluid so far away from the surface of the body that it is hardly influenced by heat transfer, i) F is called the free flow temperature, and is often written as diDC. However, when a fluid flows in a channel, (internal flow), e.g. in a heated tube, the fluid temperature at each point in a cross-section of the channel will be influenced by the heat transfer from the wall. The temperature profile for this case is shown in Figure 1.8. i) F is defined here as a cross sectional average temperature in such a way that t F is also a characteristic temperature for energy transport in the fluid along the channel axis. This definition of F links the heat flow from the wall characterised by a and the energy transported by the flowing fluid. 

Much effort has been made to study this light-off behavior of catalytic monolith. Oh and Cavendish studied the response of the monolith to a step increase in the feed stream temperature by using a onedimensional two-phase (gas and solid) model. They tracked the cross-sectional average temperature and concentration in each phase and used heat and mass transfer coefficients to describe interphase transport. The results indicated that the light-off occurs at the monolith entrance for a sufficiently high inlet exhaust temperature. For a lower inlet exhaust temperature, the light-off occurs in the downstream section, and the... 

The plug flow model heat or temperature equation is deduced from the cross section averaged temperature equation (1.299). In many tubular reactor processes the heat conduction term in the z-direction is much smaller than the heat transport by convection. For such cases the conductive transport term can be neglected and the temperature equation reduces to ... 

The heat transfer process in the entry zone between z = 0 and z = 0(aPe) is quite complex. However, far downstream (z aPe) the temperature distribution (3-211) has a relatively simple form. The cross-sectionally averaged temperature increases linearly withz... 

The objective of Taylor s analysis is to predict the evolution of the cross-sectionally averaged temperature for large times, where the length scale l c a. At this point, we thus split 0 into a cross-sectionally averaged term (9) and a remainder O, so that... 

This approximate equation governs the evolution of the cross-sectionally averaged temperature distribution in the tube. Because Pe 1 and a/if <<. 1 (this assumption must ultimately be checked), the axial conduction term in (3-227) can be neglected compared with the other two terms. We may also note that the last term in (3-227) can be written in the alternative form... 

Hence, substituting into u =), we will clearly get a term of the form const (32(0)/3z2), and thus prove that the evolution of the cross-sectionally averaged temperature distribution will occur as an effective conduction process, as asserted by Taylor. 

A remarkable property of the solution that we have derived, clearly evident in Eq. (3-244) for the cross-sectionally averaged temperature profile, is that the position in the tube where the cross-sectionally averaged temperature is maximum moves as if there were convection downstream at the mean velocity U. The temperature pulse also spreads about this plane as though there were axial conduction with an effective thermal diffusivity of... 

We can now use the Taylor dispersion equation in either of the forms (3-244) or (3-245) to show that the cross-sectionally averaged temperature profile is a Gaussian in the z direction, with the peak concentration remaining at z = 0 (i.e., converting downstream relative to fixed coordinates at the mean velocity U). 

For simplicity, the transition regions in both temperature and flow fields near the channel entrance and at the junctions of un-thermostated and thermostated regions are assumed negligibly small and thus are not considered. This implies a discontinuity in the temperature and velocity at the above junctions. In each region itself, however, there are no axial variations. Given the fact that the radial profile of liquid temperature is essentially parabolic, one can obtain the cross-sectional average temperature Tj at each region of the capillary... 

To simplify heat transfer expression, the cross sectional average temperature of the bed, T, is often used as the representative temperature. 

The x annihilation rate in a DM halo is R = nx(r)(aV)A, where nx(r) = nx,og(r) is the neutralino number density and (cross section averaged over a thermal velocity distribution at freeze-out temperature (see, e.g., Gondolo in these Proceedings). Although the annihilation cross section is a non-trivial function of the mass and physical composition of the neutralino, to our purpose it suffices to recall that the relic density is approximately given by ... 

Fig. 1.8 Temperature profile in a channel cross section. Wall temperature and average fluid temperature i F...

The flow of heat causes a change in the enthalpy flow H of the fluid. The cross sectional average fluid temperature i F is now defined such that the enthalpy flow can be written as... 

A consequence of the steepness of both functions defining the Gamow-peak is an extreme sensitivity of the Maxwellian-averaged cross section to temperature ... 

It is important to note that the important difference between the cross section averaged ID axial dispersion model equations (discussed in the previous section) and the simplified 2D model equations (presented above) is that the latter is valid locally at each point within the reactor, whereas the averaged one simply gives a cross sectional average description of the axial composition and temperature profiles. 

Here, we have written the Poiseuille solution in terms of the cross-sectional average of the velocity (uz) = U. From (3-44) and (3-45), we can see that this is just one-half of the maximum velocity that occurs at the centerline of the tube. The parameter /< that appears is the thermal diffusivity, k = k/pCp. We seek to solve (3-194) subject to the conditions that the heat flux at the tube walls is q for z > 0 and that the temperature is bounded at the centerline, r = 0. The condition on the heat flux is applied in the form (2-119). Hence... 

Furthermore, the cross sectional average of the i and 9i outlet temperatures and the average of q over the heat transfer area are needed. All these manipulations show that the temperature distributions and heat transfer in cross-flow heat exchanger are mathematically more complicated than those in parallel-and counter-flow heat exchangers. Because of this fact, practical problems associated with cross flow and other heat exchangers of similar complexity are handled by an approximation of mathematical results such as Eqs. (7.26) and (7-27) or by simpler results obtained from approximate formulations. These results are usually expressed in terms of a correction factor relative to the counter-flow (which is the most efficient) heat exchanger. ... 

Experimentally, the quantity of interest is the cross section averaged over the rotational distribution in J values at gas temperature Tg... 

Pritt and Coombe and Donovan, Fotakis, and Golde have extended the ideas behind the Ewing theory to cover the case of quadrupole-dipole attractive forces, which vary as R . This is exactly the type of force for which Sharma and Brau originally performed the perturbation theory calculation. For small values of wb/v the cross-section averaged over temperature and impact parameter is given by ... 

The integral in Eq. (3.11) represents the average reaction cross section for temperature T ... 

It is very common in reactors to have flow predominantly in one direction, say z (e.g., think of tubular reactors). The major gradients then occur in that direction, under isothermal conditions at least. For many cases then, the cross-sectional average values of concentration (or conversion) and temperature might be used in the equations instead of radial point values. The former are obtained from ... 

A proper analysis of the tine dependent behavior of.a reactor operating on thermal neutrons must take into account the important effects on its criticality, reactivity, and stability which arise from such factors as fission i products of high thermal-neutron capture cross-section, depletion, temperature, average neutron lifetime in the reactor, flux level, and reactor period. As has been seen in the requirements placed on the.reactor, considerable excess reactivity must be built into the active core before start-up. The control rods must keep the reactivity below the critical value before and during start-up. 

The relaxation constant Ki depends on the total cross section averaged over the thermal velocity distribution and on the temperature T. Integration of (8.21a-8.21b) gives... 

Since the lateral temperature variatiOTi is typically very small in free air-cooled electrokinetic flow (while the temperature gradients may be very large), it is acceptable to replace the local liquid temperature T with its cross-sectional average, T. Neglecting the effect of transverse liquid flow and averaging each term in Eq. 1 over the capillary cross-section yield... 

The first term on the right hand side is the plug-like electroosmotic velocity, and the second term is the pressure-driven flow that is slightly different from a parabola due to the small quartic term caused by radial temperature gradients. Therefore, the cross-sectional average liquid velocity in each region % is easily derived as... 

The axial Nusselt number, based on the local surface temperamre and fluid temperature cross-sectionally averaged, was investigated as well. The use of nanofluids greatly decreases both the channel wall temperature and fluid temperature... 

Figure 3.8 Cross-sectional average concentration (a) and temperature (b) profiles according to a pseudohomogeneous model (full lines two-dimensional dashed lines one-dimensional). After Ref [133] with tj = 160 C and Tw = 100 C.

Figure 3.10 Comparison of the axial temperature (a) and methanol concentration (b) according to several models (cross-sectional averages for 2D models) for a flxed-bed reactor to produce formaldehyde from methanol [135]. (Source Reproduced with kind permission of lordanidis.)...

---