Laminar fluid flow, energy balance

Reconsider steady laminar flow of a fluid m a circular tube of radius R. The fluid properties p, k, and Cp are constant, and the work done by viscous forces is negligible. The fluid flows along the.r-axis with velocity n. Tlie flow is fully developed so that i< is independent of, v and thus u = n(r). Noting that energy is transferred by mass in the A-direction, and by conduction in the r direction (heat conduction in the. v-direction.is assumed to be negligible), the steady-flow energy balance for a cylindrical shell element of thickness dr and length d. can be expressed as (Fig. 8-21)... 

A simple energy balance for laminar fluid flow in a long tube leads to the equation ... 

The kinetic-energy terms of the various energy balances developed h include the velocity u, which is the bulk-mean velocity as defined by the equati u = m/pA Fluids flowing in pipes exhibit a velocity profile, as shown in Fi 7.1, which rises from zero at the wall (the no-slip condition) to a maximum the center of the pipe. The kinetic energy of a fluid in a pipe depends on actual velocity profile. For the case of laminar flow, the velocity profile parabolic, and integration across the pipe shows that the kinetic-ertergy should properly be u2. In fully developed turbulent flow, the more common in practice, the velocity across the major portion of the pipe is not far fro... 

The heating of a viscous fluid in laminar flow in a tube of radius R (diameter, D) will now be considered. Prior to the entry plane z < 0), the fluid temperature is uniform at Tf for z > 0, the temperature of the fluid will vary in both radial and axial directions as a result of heat transfer at the tube wall. A thermal energy balance will first be made on a differential fluid element to derive the basic governing equation for heat transfer. The solution of this equation for the power-law and the Bingham plastic models will then be presented. 

In this section we first discuss the two types of fluid flow that can occur laminar and turbulent flow. Also, the Reynolds number used to characterize the regimes of flow is considered. Then in Sections 2.6, 2.7, and 2.8 the overall mass balance, energy balance, and momentum balance are covered together with a number of applications. Finally, a... 

This is the mechanical-energy loss due to skin friction for the pipe in N m/kg of fluid and is part of the F term for frictional losses in the mechanical-energy-balance equation (2.7-28). This term (Pi—Pz)/ for skin friction loss is different from the (p, — Pz) term, owing to velocity head or potential head changes in Eq. (2.7-28). That part of F which arises from friction within the channel itself by laminar or turbulent flow is discussed in Sections 2.10B and in 2. IOC. The part of friction loss due to fittings (valves, elbows, etc.), bends, and the like, which sometimes constitute a large part of the friction, is discussed in Section 2.10F. Note that if Eq. (2.7-28) is applied to steady flow in a straight, horizontal tube, we obtain (pi — Pz)/p = F. 

Kinetic energy in laminar flow. In application of the total mechanical-energy balance in Eq. (2.7-28), the average kinetic energy per unit mass of fluid is needed. For fluids, this is (S2)... 

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