Reynolds’ analogy

T Tubes, turbulent, smooth tubes, Reynolds analogy... 

Figure 12.5. The Reynolds analogy — momentum, heat and mass transfer...

Air at 330 K, flowing at 10 m/s, enters a pipe of inner diameter 25 mm, maintained at 415 K. The drop of static pressure along the pipe is 80 N/m2 per metre length. Using the Reynolds analogy between heat transfer and fluid friction, estimate the air temperature 0.6 m along the pipe. 

When the mass transfer process deviates significantly from equimolecular counterdiffusion, allowance must be made for the fact that there may be a very large difference in the molar rates of transfer of the two components. Thus, in a gas absorption process, there will be no transfer of the insoluble component B across the interface and only the soluble component A will be transferred. This problem will now be considered in relation to the Reynolds Analogy. However, it gives manageable results only if physical properties such as density are taken as constant and therefore results should be applied with care. 

The original Reynolds analogy involves a number of simplifying assumptions which are justifiable only in a limited range of conditions. Thus it was assumed that fluid was transferred from outside the boundary layer to the surface without mixing with the intervening fluid, that it was brought to rest at the surface, and that thermal equilibrium was established. Various modifications have been made to this simple theory to take account of the existence of the laminar sub-layer and the buffer layer close to the surface. 

Taylor(4) and Prandtl(8 9) allowed for the existence of the laminar sub-layer but ignored the existence of the buffer layer in their treatment and assumed that the simple Reynolds analogy was applicable to the transfer of heal and momentum from the main stream to the edge of the laminar sub-layer of thickness <5. Transfer through the laminar sub-layer was then presumed to be attributable solely to molecular motion. 

Tf auH and b8s are the velocity and temperature, respectively, at the edge of the laminar sub-layer (see Figure 12.5), applying the Reynolds analogy (equation 12,99) for transfer across the turbulent region ... 

It is thus seen that by taking account of the existence of the laminar sub-layer, correction factors are introduced into the simple Reynolds analogy. 

In this region the Reynolds analogy can be applied. Equation 12.113 becomes ... 

The simple Reynolds analogy gives a relation between the friction factor R/pu] and the Stanton number for heat transfer ... 

Firstly, using the simple Reynolds analogy (equation 12.102) ... 

It is seen that the simple Reynolds analogy is far from accurate in calculating beat transfer to a liquid. 

In this case the result obtained using the Reynolds analogy agrees much more closely with the other three methods. 

Derive an expression relating the pressure drop for the turbulent flow of a fluid in a pipe to the heat transfer coefficient at the walls on the basis of the simple Reynolds analogy. Indicate the assumptions which are made and the conditions under which you would expect it to apply closely. Air at 320 K and atmospheric pressure is flowing through a smooth pipe of 50 mm internal diameter, and the pressure drop over a 4 m length is found to be 150 mm water gauge. By how much would you expect the air temperature to fall over the first metre if the. wall temperature there is 290 K ... 

Obtain an expression for the simple Reynolds analogy between heat transfer and friction. Indicate the assumptions which are made in the derivation and the conditions under which you would expect the relation to be applicable. 

Obtain the Taylor-Prandtl modification of the Reynolds Analogy for momentum and heat transfer, and give the corresponding relation for mass transfer (no bulk flow). 

Obtain the Taylor-Prandtl modification of the Reynolds analogy between momentum and heat transfer and write down the corresponding analogy for mass transfer. For a particular system, a mass transfer coefficient of 8,71 x 10 8 m/s and a heat transfer coefficient of 2730 W/m2 K were measured for similar flow conditions. Calculate the ratio of the velocity in the fluid where the laminar sub layer terminates, to the stream velocity. 

Derive the Taylor-Prandtl modification of the Reynolds analogy between heat and momentum transfer and express it in a form in which it is applicable to pipe flow. 

Obtain the Taylor-Prandtl modification of the Reynolds Analogy between momentum transfer and mass transfer (equimolecular counterdiffusion) for the turbulent flow of a fluid over a surface. Write down the corresponding analogy for heat transfer. State clearly the assumptions which are made. For turbulent flow over a surface, the film heat transfer coefficient for the fluid is found to be 4 kW/m2 K. What would the corresponding value of the mass transfer coefficient be. given the following physical properties ... 

By using the simple Reynolds Analogy, obtain the relation between the heat transfer coefficient and the mass transfer coefficient for the gas phase for the absorption of a soluble component from a mixture of gases. If the heat transfer coefficient is 100 W/m2 K, what will the mass transfer coefficient be for a gas of specific heat capacity Cp of 1.5 kJ/kg K and density 1.5 kg/m- The concentration of the gas is sufficiently low for hulk flow effects to be negligible. 

The convective mass transfer coefficient hm can be obtained from correlations similar to those of heat transfer, i.e. Equation (1.12). The Nusselt number has the counterpart Sherwood number, Sh = hml/Di, and the counterpart of the Prandtl number is the Schmidt number, Sc = p/pD. Since Pr k Sc k 0.7 for combustion gases, the Lewis number, Le = Pr/Sc = k/pDcp is approximately 1, and it can be shown that hm = hc/cp. This is a convenient way to compute the mass transfer coefficient from heat transfer results. It comes from the Reynolds analogy, which shows the equivalence of heat transfer with its corresponding mass transfer configuration for Le = 1. Fire involves both simultaneous heat and mass transfer, and therefore these relationships are important to have a complete understanding of the subject. 

A direct numerical relationship between heat and momentum fluxes, as for the simple Reynolds analogy for a single phase, is not obtained in this case because of a basic and significant difference in heat transfer coefficient definitions. For singlephase flow in pipes, the mixed mean or integrated average temperature is used in... 

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