Thermal entry flow

This paper describes a finite element formulation designed to simulate polymer melt flows in which both conductive and convective heat transfer may be important, and illustrates the numerical model by means of computer experiments using Newtonian extruder drag flow and entry flow as trial problems. Fluid incompressibility is enforced by a penalty treatment of the element pressures, and the thermal convective transport is modeled by conventional Galerkin and optimal upwind treatments. 

The thermal entry flow with fully developed velocity profile... 

Laminar flow that is developing hydrodynamically or thermally, (entry flow) constant wall temperature ... 

Laminar Flow. Thermal entry length solutions with developed velocity profiles are summarized in Refs. 19 and 44 for a large number of practically important flow passage geometries with extensive comparisons. 

For internal flows, thermal boundary layers develop from both top and bottom surfaces and develop into two regions thermal entry length and thermal fully developed regions similar to hydrodynamic internal flow as shown in Figure 6.8. 

To overcome thermal entry effects, the segments may be virtually stacked with the outlet conditions from one segment that becomes the inlet conditions for the next downstream section. In this approach, axial conduction cannot be included, as there is no mechanism for energy to transport from a downstream section back to an upstream section. Thus, this method is limited to reasonably high flow rates for which axial conduction is negligible compared to the convective flow of enthalpy. At the industrial flow rates simulated, it is a common practice to neglect axial conduction entirely. The objective, however, is not to simulate a longer section of bed, but to provide a developed inlet temperature profile to the test section. 

Alternatively, reactant and product gases can be distributed to and removed from individual cells through internal pipes in a design analogous to that of filter presses. Care must be exercised to assure an even flow distribution between the entry and exit cells. The seals in internally manifolded stacks are generally not subject to electrical, thermal, and mechanical stresses, but are more numerous than in externally manifolded stacks. 

Fig. 4.16 Illustration of the Graetz problem. A fully developed parabolic velocity profile is established in a circular duct and remains unchanged over the length of the duct. There is a sudden jump in the wall temperature, and the fluid temperature is initially uniform at the upstream wall temperature. The thermal-entry problem is to determine the behavior of the temperature profile as it changes to be uniform at the downstream wall temperature. Because the flow is incompressible, the velocity distribution does not depend on the varying temperatures.

The Circular Tube Thermal-Entry-Length, with Hydrodynamically Fully Developed Laminar Flow... 

A power plant operating on heat recovered from the exhaust gases of internal-combustion < uses isobutane as the working medium in a modified Rankine cycle in which the upper pressure I is above the critical pressure of isobutane. Thus the isobutane does not undergo a change of p" as it absorbs heat prior to its entry into the turbine. Isobutane vapor is heated at 4,800 kPa to 2 and enters the turbine as a supercritical fluid at these conditions. Isentropic expansion in the turh produces superheated vapor at 450 kPa, which is cooled and condensed at constant pressure, resulting saturated liquid enters the pump for return to the heater. If the power output of the modi Rankine cycle is 1,000 kW, what is the isobutane flow rate, the heat-transfer rates in the heater condenser, and the thermal efficiency of the cycle ... 

We start this chapter with a general physical description of internal flow, and the average velocity and average temperature. We continue with the discussion of the hydrodynamic, and thermal entry lengths, developing flow, and fully developed flow. We then obtain the velocity and temperature profiles for fully developed laminar flow, and develop relations for the friction factor and Nusselt nmnber. Hinally we present empirical relations for developing and full developed flows, and demonstrate their use. 

B Have a visual understanding of different flow regions in internal flow, such as Ihe entry and the fully developed flow regions, and calculate hydrodynamic and thermal entry lengths,... 

Th regioii of flow over which the thermal boundary layer develops and re.iches (he tube center i.s called the thermal entrance region, and the length of this region is called the thermal entry length L,. Flow in the thermal... 

During laminar flow in a tube, the magnitude of the dimensionless Prandtl number Pr is a measure of the relative growth of the velocity and thermal boundary layers. For fluids with Pr = I, such as gases, the two boundary layers essentially coincide with each other. For fluids with Pr > I, such as oils, the velocity boundary layer outgrows the thermal boundary layer. As a result, the hydrodynamic entry length is smaller than the thermal entry length. The opposite is tnie for fluids with Pr < 1 such as liquid metals. 

The hydrodynamic entry length is usually taken to be the distance from the lube entrance where the wall shear. stress (and thus the fficliou factor) reaches within about 2 percent of the fully developed value. In laminar flow, the hydrodynamic and thermal entry lengths are given approximately as (see Kays and Crawford (1993) and Shah and Bhatli (1987)]... 

In turbulent flow, the intense iqjxing during random fluctuations usually overshadows the effects of molecular diffusion, and therefore the hydrodynamic and thermal entry lengths ate of about the same size and independent of the PflLndil number. The hydrodynamic entry length for turbulent flow can be detennined from [see Bbatti and Shah (1987) and 7.hi-qing (1982)]... 

The entry length is much shorter in turbulent flow, as expected, and its dependence on the Reynolds number is weaker. In many lube flotvs of practical interest, the entrance effects become insignificant beyond a tube length of 10 diameters, and the hydrodynamic and thermal entry lengths are approximately taken to be... 

C How is the thermal entry length defined for flow in a tube In what region is the flow in a tube fully developed ... 

S-18C Consider the flow of mercury (a liquid meial) in a tube. How will the hydrodynamic and thermal entry lengths compare if the flow is laminar How would they compare if (he flow uere turbulent ... 

Comparison of the thermal conductivities of prolate (p) and oblate (o) nematic liquid crystals. The entries for zero field have been obtained by using the Green-Kubo relation (3.3). The entries for finite field have been obtained by applying the heat flow algorithm (3.5). Note that the EMD GK estimates and the NEMD estimates agree within the statistical error. 

The unusual behavior of Nu decreasing with increasing Re in the laminar regime in microchannels may alter the status of thermal development and hence the conventional thermal entry length, since the variation of the heat transfer coefficient along the flow is a variation of the boundary condition. 

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