Uniform-wall-flux

Figure 4. The effect of wall cooling on susceptor heat flux uniformity.

The thermal entrance region in a hydrodynamically fully developed flow in a rectangular duct may be studied by the use of the integral method. In this section, the uniform wall temperature and the uniform wall heat flux cases are discussed. The physical model is based on the following assumptions ... 

Table 9.3 Average Nu for uniform wall heat flux and constant wall temperature, Ostrach [3].

This is, then, the temperature distribution for folly developed laminar pipe flow when the heat flux at the wall is uniform. It can be written in terms of die specified uniform wall heat flux, qw, by noting that when Eq. (4.30) is used to give the value of dTtdr r ro in Eq. (4.22), the following is obtained ... 

This can be written in termsj of the specified uniform wall heat flux, qw> by again noting that because qjk = dTldr rmrg, Eq. ( ) gives ... 

In the uniform wall heat flux case, the definition of the dimensionless temperature is such that ... 

Nusselt number and center line temperature variation in developing flow in a pipe with a uniform wall heat flux. 

The computer program DEVPIPE discussed above allows either a uniform wall heat flux or a uniform wall temperature to be considered. The variations of Nud, Nuomi and the dimensionless center line temperature, 6, with Z given by this program for the uniform wall heat flux case are shown in Fig. 4.19. 

A solution for the uniform wall heat flux case can also be obtained using the separation of variables approach discussed above. 

The Modifications to the above analysis to deal with the uniform wall heat flux case are basically the same as that required for pipe flow and will not be discussed here. 

Thus, the only difference required when the wall heat flux is specified is that instead of setting 0,n = 1, Eq. (4.246) is used to determine 0,. TTie computer program discussed above allows for either a uniform wall temperature or a uniform wall heat flux. 

Consider fully developed laminar flow fluid through a circular pipe with a uniform wall heat flux. If heat is generated uniformly in the fluid, perhaps as the result of a chemical reaction, at a rate of q per unit volume, determine the value of the Nusselt number based on the difference between the wall temperature and the mean fluid temperature in the pipe. 

In some situations it is possible to find the heat transfer rate with adequate accuracy by assuming that the velocity is constant across the duct. Le to assume that so-called slug flow exists. Find the temperature distribution and the Nusselt number in sllug flow in a plane duct when the thermal field is fully developed and when there is a uniform wall heat flux. 

Tbe numerical procedure for solving the laminar boundary layer equations for forced convection that was described in Chapter 3 is easily extended to deal with combined convection. The details of the procedure are basically the same as those for forced convection and the details will not be repeated here [16]. A computer program, LAMBMIX, based on the procedure is available in the way discussed in the Preface. This program can actually allow the wall temperature or wall heat dux to vary with X but as available, the program is set for the case of a uniform wall temperature or a uniform wall heat flux. 

The variations of NuJ jRex with Gx for the uniform surface temperature case and of NuxljRex with G for the uniform wall heat flux case for various values of Pr as given by the computer program are shown in Figs. 9.9 and 9.10, respectively. 

Abu Mulaweh. H.L, Armaly, B.F., and Chen, T.S., Measurements of Laminar Mixed Convection Adjacent to a Vertical Plate-Uniform Wall Heat Flux Case , J. Heat Transfer. Vol. 114, Nov. 92, pp. 1057-1059,1992. 

The wall temperature gradient is now related to the specified uniform wall heat flux qw by substituting q. (10,155) into Eq. (10.144). This gives ... 

For laminar flow (ReD < 2100) that is fully developed, both hydro-dynamically and thermally, the Nusselt number has a constant value. For a uniform wall temperature, NuD = 3.66. For a uniform heat flux through the tube wall, NuD = 4.36. In both cases, the thermal conductivity of the fluid in NuD is evaluated at Tb. The distance x required for a fully developed laminar velocity profile is given by [(x/D)/ReD] 0.05. The distance x required for fully developed velocity and thermal profiles is obtained from [(x/D)/(ReD Pr)] = 0.05. 

SOLUTION A liquid mixture flowing in a tube is subjected to uniform wall heat flux. The friction coefficients are to be determined for the bell-mouth and square-edged inlet cases. 

Ethylene glycol-distilled water mixture with a rhass fraction of 0.6 and a flow I rate of 2.5 x 10 m /s flows inside a tube with an inside diameter of 0.0158 m subjected to uniform wall heat flux. For this flow, determine the Nusselt number at the location x/D = 90 if the inlet configuration of the tube ] iS (a) re-entrant, (b) square-edged, and (c) beli-mouth. At this location, the local Grashof number is Gr = 51,770. The properties of ethylene glycol-distillcd water mixture at the location of interest are Pr = 29.2, v = 3.12 x 10 m% and P(//is = 1.77.. 

A. J. Ghajar and L. M. Tam. "Flow Regime Map for a Horizontal Pipe with Uniform Wall Heat Flux and Three Inlet Configurations. Experimental Thermal and Fluid Science, Vol. 10 (1995), pp. 287—297. 

The Nusselt number (Nu) for laminar, fully developed flow is eonsfanf and independenf of Re, Pr, and fhe axial loeafion. Under fhese flow eondifions Nu values for a eylindrieal ehannel wifh uniform wall heaf flux and uniform wall femperature are 4.36 and 3.66, respeefively. More Nusself number values for reefangular ehannels are available in Table 3. 

For the ease of the uniform wall heat flux boundary eondition, we again apply the integral transform teehnique using a method similar to the eylindrieal solution. The Nu number is given as ... 

Boundary Layer Theory. The Reynolds number for flow-through hollow fibers during our experiments was at most about 0.02 cm (diameter) x 4 cm/sec (velocity) x 1.0 g/cm (density)/ 0.007 poise (viscosity) 11 therefore, a boundary layer theory is needed for laminar flow in tubes. Because of its simplicity, the most attractive available theory is an approximate result of thln-film theory. This theory is restricted to a description of boundary layers that are thin in comparison to the tube radius. Furthermore, the ultrafiltrate velocity, J, must not vary along the tube length (uniform-wall-flux theory). At the centerline or axis of the fiber, the impermeable solute concentration C = C... 

For rejection of tracer solutes out of saline solutions, the uniform-wall-flux restriction is not a problem, because the axial pressure drop is generally less than 10% of the average transmem-... 

Figure 2. Comparison of equations for uniform wall flux boundary layer modulus...

Analytic expressions for the protein osmotic pressure were needed to test the applicability of uniform wall flux theory. Vilker s data (3) at pH 7.4 were used for BSA, interpolated over our range of interest and converted to our concentration units (g/1 of olutlon) by using a protein partial specific volume of... 

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