Analogy between heat transfer and fluid friction

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. 

As we shall see in Chap. 6, Eq. (5-116 ) predicts heat-t rar.sfcr coefficients that are somewhat higher than those observed in experiments. The purpose of the discussion at this point has been to show that one may arrive at a relation for turbulent heat transfer in a fairly simple analytical fashion. As we have indicated earlier, a rigorous development of the Reynolds analogy between heat transfer and fluid friction involves considerations beyond the scope of our discussion and the simple path of reasoning chosen here is offered for the purpose of indicating the general nature of the physical processes. 

The analogy between heat transfer and fluid friction [Eq. (5-56)] may also be used when the friction coefficient is known. Summarizing the relations used for high-speed heat-transfer calculations ... 

The drag coefficient for a sphere at Reynolds numbers less than 100 may be approximated by CD - b Re 1, where b is a constant. Assuming that the Colburn analogy between heat transfer and fluid friction applies, derive an expression for the heat loss from a sphere of diameter d and temperature T released from rest and allowed to fall in a fluid of temperature T.. (Obtain an expression for the heat lost between the time the sphere is released and the time it reaches some velocity v. Assume that the Reynolds number is less than 100 during this time and that the sphere remains at a constant temperature.)... 

Equation (12.52) is a statement of the Colburn analogy between heat transfer and fluid friction. The factor defined as is called the... 

Analogies have been developed between heat transfer and fluid friction (i.e., energy and momentum analogies) [5-7]. Similarly, analogies have been developed for systems having a transfer of matter by diffusion accompanied by heat transfer [8-10], Our discussion here is limited to flow in a tube. 

Low-Speed. Flow. Heat transfer is best found from the Reynolds analogy, the relationship between heat transfer and skin friction evaluated through analyses utilizing the empirical velocity distributions cited earlier. Knowledge of the flow field, which is independent of the temperature field when the fluid properties are constant, can be used directly to define the temperature field for a variety of thermal conditions and to evaluate the resulting convective heat transfer rates. For low-speed, constant-property flow, the energy equation is... 

Metzner and Friend [73] measured turbulent heat transfer rates with aqueous solutions of Carbopol, corn syrup, and slurries of Attagel in circular-tube flow. They developed a semi-theoretical correlation to predict the Stanton number for purely viscous fluids as a function of the friction factor and Prandtl number, applying Reichardt s general formulation for the analogy between heat and momentum transfer in turbulent flow ... 

Lin, Hsiao Tsung and Chen, Yao Han, The Analogy Between Fluid Friction and Heat Transfer of Laminar Mixed Convection Flat Plates , lnt. J. Heat and Mass Transfer, Vol. 37, Nd 11, pp. 1683-1686, 1994. 

Equation (5-56), called the Reynolds-Colburn analogy, expresses the relation between fluid friction and heat transfer for laminar flow on a flat plate. The heat-transfer coefficient thus could be determined by making measurements of the frictional drag on a plate under conditions in which no heat transfer is involved. 

For the flow system in Example 5-4 compute the drag force exerted on the first 40 cm of the plate using the analogy between fluid friction and heat transfer. 

Various analyses, similar to the one for the universal velocity profile above, have been performed to predict turbulent-boundary-layer heat transfer. The analyses have met with good success, but for our purposes the Colburn analogy between fluid friction and heat transfer is easier to apply and yields results which are in agreement with experiment and of simpler form. 

Calculate the drag (viscous-friction) force on the plate in Prob. 5-21 under the conditions of no heat transfer. Do not use the analogy between fluid friction and heat transfer for this calculation i.e., calculate the drag directly by evaluating the viscous-shear stress at the wall. 

The empirical correlations presented above, with the exception of Eq. (6-7), apply to smooth tubes. Correlations are, in general, rather sparse where rough tubes are concerned, and it is sometimes appropriate that the Reynolds analogy between fluid friction and heat transfer be used to effect a solution under these circumstances. Expressed in terms of the Stanton number,... 

The mechanism of heat flow in forced convection outside tubes differs from that of flow inside tubes, because of differences in the fluid-flow mechanism. As has been shown on pages 59 and 106 no form drag exists inside tubes except perhaps for a short distance at the entrance end, and all friction is wall friction. Because of the lack of form friction, there is no variation in the local heat transfer at different points in a given circumference, and a close analogy exists between friction and heat transfer. An increase in heat transfer is obtainable at the expense of added friction simply by increasing the fluid velocity. Also, a sharp distinction exists between laminar and turbulent flow, which calls for different treatment of heat-transfer relations for the two flow regimes. 

T. von Karmen, The Analogy Between Fluid Friction and Heat Transfer, Trans ASME, (61) 705-710,1939. 

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