Heat laminar pipe flow

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 ... 

T. F. Lin, K. H. Hawks, and W. Leidenfrost, Analysis of Viscous Dissipation Effect on Thermal Entrance Heat Transfer in Laminar Pipe Flows with Convective Boundary Conditions, Wiirme-und Stoffubertragung, (17) 97-105,1983. 

J. C. Kuo, and T. F. Lin, Steady Conjugate Heat Transfer in Fully Developed Laminar Pipe Flows, J. of Thermophysics and Heat Transfer, (2/3) 281-283,1988. 

FIGURE 10.6 Experimental results for laminar pipe flow heat transfer for constant wall heat flux boundary conditions [35]. 

In Chapter 1, we considered the problem of heat transfer in laminar pipe flow. If it is permissible to ignore axial conduction, the transport equation we derived there was given by Eq. 1.31 with the term kd T/dz 0... 

The modeling of cooling a fluid in a laminar pipe flow was considered in Example 10.3. Neglecting the axial heat conduction relative to the convection term, the following heat balance equation describes the temperature change inside the pipe... 

Here, h is the enthalpy per unit mass, h = u + p/. The shaft work per unit of mass flowing through the control volume is 6W5 = W, /m. Similarly, is the heat input rate per unit of mass. The fac tor Ot is the ratio of the cross-sectional area average of the cube of the velocity to the cube of the average velocity. For a uniform velocity profile, Ot = 1. In turbulent flow, Ot is usually assumed to equal unity in turbulent pipe flow, it is typically about 1.07. For laminar flow in a circiilar pipe with a parabohc velocity profile, Ot = 2. 

All pipe-line work to date has dealt with fluids which are not thixotropic and rheopectic. To an extent this may be justified because the limiting conditions (at startup—for thixotropic materials, and after long times of shear for rheopectic fluids) in pipe flow and some mixing problems are of primary importance. Design for these conditions would be similar to the techniques discussed herein for other fluids. This is not true of problems in heat transfer, however, and inception of work on the laminar flow of thixotropic fluids in round pipes would appear to be in order as a prerequisite to an understanding of such more complex nonisothermal problems. 

Internal flows of the type here being considered occur in heat exchangers, for example, where the fluid may flow through pipes or between closely spaced plates that effectively form a duct Although laminar duct flows do not occur as extensively as turbulent duct flows, they do occur in a number of important situations in which the size of the duct involved is small or in which the fluid involved has a relatively high viscosity. For example, in an oil cooler the flow is usually laminar. Conventionally, it is usual to assume that a higher heat transfer rate is achieved with turbulent flow than with laminar flow. However, when the restraints on possible solutions to a particular problem are carefully considered, it often turns out that a design that involves laminar flow is the most efficient from a heat transfer viewpoint. 

Nusselt s film condensation theory presumes a laminar film flow. As the amount of condensate increases downstream, the Reynolds number formed with the film thickness increases. The initially flat film becomes wavy and is eventually transformed from a laminar to a turbulent film the heat transfer is significantly better than in the laminar film. The heat transfer in turbulent film condensation was first calculated approximately by Grigull [4.14], who applied the Prandtl analogy for pipe flow to the turbulent condensate film. In addition to the quantities for laminar film condensation the Prandtl number appears as a new parameter. The results can not be represented explicitly. In order to obtain a clear representation, we will now define the Reynolds number of the condensate film... 

In Table 5.44, it can be seen that the pitch of the helicoidal coil has almost no influence on the Nusselt number. However, the studies by Yang et al. [206, 207] have shown a positive effect of the pitch on the Nusselt number when Pr > 1. In addition, the experiments conducted by Austen and Soliman [208] indicated that the Nusselt number for the laminar flow of water (3 < Pr < 6) in the uniformly heated helicoidal pipe is in good agreement with the prediction from Manlapaz and Churchill [201]. 

J. W. Ou, and K. C. Cheng, Viscous Dissipation Effects on Thermal Entrance Heat Transfer in Laminar and Turbulent Pipe Flows with Uniform Wall Temperature, AlAA, paper no. 74-743 or ASME paper no. 74-HT-50,1974. 

Y. S. Kadaner, Y. P. Rassadkin, and E. L. Spektor, Heat Transfer in Laminar Liquid Flow through a Pipe Cooled by Radiation, Heat Transfer-Sov. Res., (3/5) 182-188,1971. 

An exception to the generally observed drag reduction in turbulent channel flow of aqueous polymer solutions occurs in the case of aqueous solutions of polyacrylic acid (Carbopol, from B.F. Goodrich Co.). Rheological measurements taken on an oscillatory viscometer clearly demonstrate that such solutions are viscoelastic. This is also supported by the laminar flow behavior shown in Fig. 10.20. Nevertheless, the pressure drop and heat transfer behavior of neutralized aqueous Carbopol solutions in turbulent pipe flow reveals little reduction in either of these quantities. Rather, these solutions behave like clay slurries and they have been often identified as purely viscous nonnewtonian fluids. The measured dimensionless friction factors for the turbulent channel flow of aqueous Carbopol solutions are in agreement with the values found for clay slurries and may be correlated by Eq. 10.65 or 10.66. The turbulent flow heat transfer behavior of Carbopol solutions is also found to be in good agreement with the results found for clay slurries and may be calculated from Eq. 10.67 or 10.68. 

Even when the flow in the body of the boundary layer is turbulent, flow remains laminar in the thin layer close to the solid smface, the so-called laminar sub-layer. Indeed, the bulk of the resistance to momentum, heat and mass transfer hes in this thin film and therefore interphase heat and mass transfer rates may be increased by decreasing its thickness. As in pipe flow, the laminar sub-layer and the turbulent region are separated by a buffer layer in which viscous and inertial effects are of comparable magnitudes, as shown schematically in Figure 7.1. 

Certainly, the most important convective heat-transfer process industrially is that of cooling or heating a fluid flowing inside a closed circular conduit or pipe. Diflferent types of correlations for the convective coefficient are needed for laminar flow below 2100), for fully turbulent flow above 6000), and for the transition region (/Vr between 2100 and 6000). 

The heat-transfer phenomena for forced convection over exterior surfaces are closely related to the nature of the flow. The heat transfer in flow over tube bundles depends largely on the flow pattern and the degree of turbulence, which in turn are functions of the velocity of the fluid and the size and arrangement of the tubes. The equations available for the calculation of heat transfer coefficients in flow over tube banks are based entirely on experimental data because the flow Is too complex to be treated analytically. Experiments have shown that, in flow over staggered tube banks, the transition from laminar to turbulent flow Is more gradual than in flow through a pipe, whereas for in-line tube bundles the transition phenomena resemble those observed in pipe flow. In either case the transition from laminar to turbulent flow begins at a Reynolds number based on the velocity in the minimum flow area of about 100, and the flow becomes fully turbulent at a Reynolds number of about 3,000. The equation below can be used to predict heat transfer for flow across ideal tube banks. 

The steady laminar flow of a liquid through a heated cylindrical pipe has a parabolic velocity profile if natural convection effects, and variation of physical properties with temperature are neglected [4], If the fluid entering the heated section is at a uniform temperature (Ti) and the wall is maintained at a crmstant temperature (T ), develop Graetz s solution by neglecting the thermal conductivity in the axial directiOTi. 

In a recent book by Molerus and Wirth (1997), the recommended heat transfer correlations for packed beds can be summarized as follows. For fully developed laminar flow, an approximation formula for the mean Nusselt number, derived from the pipe flow analogy, was proposed as... 

Special solutions exist for porous tubes/pipes for laminar channel flow with no heat transfer for non-Newtonian flow with small uniform suction or injection (Narasimhan, 1961), for swirling flow with accelerating walls with small uniform suction (Banks and Zaturska, 1996), for pulsating flow (Chang et al., 1989), and for expanding walls with arbitrary uniform suction (Si et al., 2011a,b) or arbitrary uniform injection (Goto and Uchida, 1980). 

Derive the heat transfer counterpart for laminar entry flow and for turbulent flow in a cylindrical pipe. 

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