Heat transfer in laminar flow

In the flow of a fluid past a phase boundary, such as that through which mass transfer occurs, there will be a velocity gradient within the fluid, which results in a transfer of momentum through the fluid. In some cases there is also a transfer of heat by virtue of a temperature gradient. The processes of mass, momentum, and heal transfer under these conditions are intimately related, and it is useful to consider this point briefly. 

When a temperature gradient exists between the fluid and the plate, the rate of heat transfer in the laminar region of Fig. 2.6 is 

In a gas at relatively low pressure the heat energy is transferred from one position to another by the molecules traveling from one layer to another at a lower temperature. A simplified kinetic theoiy leads to the expression 

Equation (2.49) and (2.52) would give the dimensionless ratio v/a C p/k equal to Cp/C . A more advanced kinetic theory modifies the sizo of the ratio, known as the Prandtl number Pr, and experimentally it has the range 0.65 to 0.9 

The third dimensionless group, formed by dividing the thermal by the mass diffusivity, is the Lewis number, Le = a/Z) = Sc/Pr, and it plays an important part in problems of simultaneous heat and mass transfer, as will be developed later. 

Statement of the problem. Thermal boundary layer. Let us consider heat transfer to a flat plate in a longitudinal translational flow of a viscous incompressible fluid with velocity U at high Reynolds numbers. We assume that the temperature on the plate surface and remote from it is equal to the constants Ts and 7], respectively. The origin of the rectangular coordinates X, Y is at the front edge of the plate, the X-axis is tangent, and the Y-axis is normal to the plate. 

Numerous experiments and numerical calculations show that the laminar hydrodynamic boundary layer occurs for 5 x 102 Rex 5 x 105 to 106 [427]. In this region the thermal Peclet number Pe = Pr Rex is large for gases and common liquids. For liquid metals, there is a range of Reynolds numbers, 104 Rex 106, where the Peclet numbers are also large. 

Taking into account the previous discussion, we restrict ourselves to the case of high Peclet numbers, for which the longitudinal molecular heat transfer may be neglected. The corresponding equations of the thermal boundary layer and the boundary conditions have the form 

Mass Transfer in Films, Tubes, and Boundary Layers 

Here the dimensionless variables are introduced by formulas (3.1.32) with the characteristic length a = v/Ui and v is the kinematic viscosity of the fluid. 

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Celata GP, Cumo M, Marcom V, McPhail SJ, Zummo Z (2006) Micro-tube liquid single phase heat transfer in laminar flow. Int. J. Heat Mass Transfer 49 3538-3546... 

Heat transfer in laminar flow on a vertical wall... 

The constant B in Equation (27) corresponds to the asymptotic Sherwood number for constant reactant concentration at the wall, which is the same as the asymptotic Nusselt number, which characterizes the heat transfer in laminar flow at a constant wall temperature. The constant B depends on the geometry of the channel values are summarized in Table 1. 

Mizushina, T. and Kurivaki, Yu., Heat transfer in laminar flow of pseudoplastic fluids in a circular tube. In Heat and Mass Transfer, Vol. 3, Minsk, 1968 [in Russian]. 

R.O.C. Guedes, R.M. Cotta, and N.C.L. Bram, Conjugated Heat Transfer in Laminar Flow Between Parallel - Plates Channel, 10th Brazilian Congress of Mechanical Engineering, Rio de Janeiro, Brazil, 1989. 

Scale Size Range, m Typical Flow Regime Molecular Diffusion in Laminar Flow Heat Transfer in Laminar Flow... 

In this section three types of heat transfer in laminar flow are considered ... 

EFF OF NATURAL CONVECTION IN LAMINAR-FLOW HEAT TRANSFER In laminar flow at low velocities, in large pipes, and at large temperature drops, natural convection may occur to such an extent that the usual equations for laminar-flow heat transfer must be modified. The effect of natural convection in tubes is found almost entirely in laminar flow, as the higher velocities characteristic of flow in the transition and turbulent regimes overcome the relatively gentle currents of natural convection. 

Difficult heat-exchange problems occur when one of two fluid streams has a much lower heat-transfer coefficient than the other, A typical case is heating a fixed gas, such as air, by means of condensing steam. The individual coefficient for the steam is typically 100 to 200 times that for the airstream consequently, the overall coefficient is essentially equal to the individual coefiBcient for the air, the capacity of a unit area of heating surface will be low, and many meters or feet of tube will be required to provide reasonable capacity. Other variations of the same problem are found in heating or cooling viscous liquids or in treating a stream of fluid at low flow rate, because of the low rate of heat transfer in laminar flow. 

Laminar Flow. The Graetz or Leveque solutions25 26 for convective heat transfer in laminar flow channels, suitably modified for mass transfer, may be used to evaluate the mass transfer coefficient where the laminar parabolic velocity profile is assumed to be established at the channel entrance but where the concentration profile is under development down the full length of the channel. For all thin-channel lengths of practical interest, this solution is valid. Leveque s solution26 gives ... 

For the heat transfer in laminar flow in coiled annular ducts, Garimella et al. [233] experimentally obtained the following correlation to calculate the heat transfer coefficient ... 

This equation indeed shows that the Dean number represents the heat transfer in laminar flow the coil ratio (dQ - d,) D is another factor to affect the heat transfer. 

M. H. Hu, and Y. P. Chang, Optimization of Finned Tubes for Heat Transfer in Laminar Flow, J. Heat Transfer, (95) 332-338,1973 for numerical results, see M. H. Hu, Flow and Thermal Analysis for Mechanically Enhanced Heat Transfer Tubes, Ph.D. thesis, State University of New York at Buffalo, 1973. 

There are more data for heat transfer in laminar flow than for mass transfer, and the correlations should be similar, with Pr and Nu replacing Sc and Sh. An empirical equation for heat transfer at Graetz numbers greater than 20 is [15]... 

In the transition region for a between 2100 and 6000, the empirical equations are not well defined just as in the case of fluid friction factors. No simple equation exists for accomplishing a smooth transition from heat transfer in laminar flow to turbulent flow, i.e., a transition from Eq. (4.5-4) at a = 2100 to Eq. (4.5-8) at a T/r = 6000. 

Laminar Flow Although heat-transfer coefficients for laminar flow are considerably smaller than for turbulent flow, it is sometimes necessary to accept lower heat transfer in order to reduce pumping costs. The heat-flow mechanism in purely laminar flow is conduction. The rate of heat flow between the walls of the conduit and the fluid flowing in it can be obtained analytically. But to obtain a solution it is necessary to know or assume the velocity distribution in the conduit. In fully developed laminar flow without heat transfer, the velocity distribution at any cross section has the shape of a parabola. The velocity profile in laminar flow usually becomes fully established much more rapidly than the temperature profile. Heat-transfer equations based on the assumption of a parabolic velocity distribution will therefore not introduce serious errors for viscous fluids flowing in long ducts, if they are modified to account for effects caused by the variation of the viscosity due to the temperature gradient. The equation below can be used to predict heat transfer in laminar flow. 

Shrestha, G.M., 1968. Heat transfer in laminar flow in a uniformly porous channel with an applied transverse magnetic field. Appl. Sci. Res. 19, 352-369. 

Terrill, R.M., 1965b. Heat transfer in laminar flow between parallel porous plates. Int. J. Heat Mass Transf. 8, 1491-1497. 

The value for the asymptotic Sh number for constant concentration at the wall is identical to the asymptotic Nusselt number Nu o characterizing the heat transfer in laminar flow at constant wall temperature. The values for NUoo and Shoo depend on the geometry of the chaimel as summarized in Table 2.1. 

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