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Heat transfer in laminar flow

With laminar pattern of flow heat is transferred from one fluid layer to another in direction normal to the wall by conduction. At the same time each layer has own longitudinal velocity. Thus we have longitudinal convective heat transfer together with conduction. Let us consider course of heat transfer process along the tube (figure 4.17).

On fluid motion along the tube layers near the wall are heated or cooled. In the initial part of the tube central fluid core has temperature equal to the inlet one. This core does not take part in heat exchange. All the temperature change is concentrated in the layers near the wall. So thermal boundary layer is formed in the initial tube part near the surface and its thickness gradually increases on getting farther from the inlet. At a certain distance from the inlet equal l_in.t thermal layers join each other.

Tube section with the length l _in.t is called thermal stabilization section or initial thermal section.

Let fluid temperature be constant in the initial tube section and different from that of the wall. On motion heat exchange between the fluid and the wall goes and fluid temperature changes. Near the wall surface thermal boundary layer is formed and its thickness is gradually increased in direction of flow. At a certain distance from tube initial section l_in.t thermal layers join each other and all the fluid flux takes part in heat exchange.

Distance l _in.t can be approximately found like

.

Within the thermal initial stabilization section temperature gradient in the fluid near the wall diminishes sharper on getting father from the inlet than temperature drop because central part of the flux does not take part in heat exchange yet. Thus it can be concluded from heat transfer equation that

local value of heat transfer coefficient a_loc gradually diminishes along the tube. This decrease goes until thermal boundary layers join each other. Then both of the values and diminish with the same speed and local value of heat transfer coefficient is constant. On figure 4.18 the change of heat transfer coefficient local value along the tube is given.

This graph shows that distance on which average heat transfer coefficient is stabilized is always more than the same value for local heat transfer coefficient.

Micheev M.A. proposed to use following criteria equation for determining heat transfer coefficient with laminar pattern of flow

This equation can be used with Rе_f>10 and value of Pr_f>0,06 and ratio l/d >10.

It is rather difficult to account for free convection influence with tube different positions and different heating-cooling conditions. Approximate estimations for heat transfer coefficient with viscous-gravity flow can be made using following expression

.

In this expression average temperature of fluid in the tube is used as characteristic. Tube inside diameter is considered to be characteristic dimension. Coefficient e _f accounts for the change of average heat transfer coefficient along the tube. If then e _f =1, with l/d< 50 factor e_f can be approximately estimated with the use of Table 4.1.

Table 4.1 –Values of factor e _f with laminar pattern of flow.

Viscous-gravity flow exists with Gr^.Pr>8^.10⁵, where Gr= gbDtd ³/n²; Dt =|(t _w- t ₀)|; t ₀ –fluid temperature at the tube inlet; physical parameters entering Gr and Pr are chosen at temperature t =0,5(t ₀+ t _w)

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