Energy Balance of a Flowing Fluid

ENERGY BALANCE OF A FLOWING FLUID The energy terms associated with the flow of a fluid are 

One kind of application of this equation is to the filling and emptying of vessels, of which Example 6.2 is an instance. 

Unsteady Flow of an Ideal Gas through a Vessel An ideal gas at 350 K is pumped into a 1000 L vessel at the rate of 6 g mol/min and leaves it at the rate of 4 g mol/min. Initially the vessel is at 31 OK and 1 atm. Changes in velocity and elevation are negligible. The contents of the vessel are uniform. There is no work transfer. 

The temperature will be found as a function of time d with both /i = 15 and li = 0. 

Friction is introduced into the energy balance by noting that it is a mechanical process, dWf, whose effect is the same as that of an equivalent amount of heat transfer dQf. Moreover, the total effective heat transfer results in a change in entropy of the flowing liquid given by 

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Work of a flowing fluid against pressure. One additional flow of energy for systems open to the flow of mass must be included in the energy balance equation it is more subtle than the energy flows just considered. This is the energy flow that arises from the fact that as an element of fluid moves, it does work on the fluid ahead of it, Snd the fluid behind it does work on it. Clearly, each of these work terms is of the PAV type. To evaluate this energy flow term, which occurs only in systems open to the flow of mass, we. will compute the net work done as one fluid element of mass (M)i enters a system, such as the valve in Fig. 3.1-1, and another fluid element of... 

In practice, the loss term AF is usually not deterrnined by detailed examination of the flow field. Instead, the momentum and mass balances are employed to determine the pressure and velocity changes these are substituted into the mechanical energy equation and AFis deterrnined by difference. Eor the sudden expansion of a turbulent fluid depicted in Eigure 21b, which deflvers no work to the surroundings, appHcation of equations 49, 60, and 68 yields... 

Example 3 Venturi Flowmeter An incompressible fluid flows through the venturi flowmeter in Fig. 6-7. An equation is needed to relate the flow rate Q to the pressure drop measured by the manometer. This problem can he solved using the mechanical energy balance. In a well-made venturi, viscous losses are neghgihle, the pressure drop is entirely the result of acceleration into the throat, and the flow rate predicted neglecting losses is quite accurate. The inlet area is A and the throat area is a. 

In general the flow of a pure fluid is described by the equation of continuity, the three equations of motion, and the equation of energy balance. In addition, one has to specify boundary and initial conditions and also the dependence of p on p and T (the thermal equation of state) and the dependence of Cv or U on p and T (the caloric equation of state). 

Figure 6.1. Energy balances on fluids in completely mixed and plug flow vessels, (a) Energy balance on a bounded space with uniform conditions throughout, with differential flow quantities dmi and dm2. (b) Differential energy balance on a fluid in plug flow in a tube of unit cross section.

The energy equation may be derived using the first law of thermodynamics for a differential volume element in a flow field. In the absence of radiation and heat sources or sinks in the fluid, the energy balance on a differential volume element AxAyAz about a point (x,y,z) may be expressed as... 

For any given operating conditions involving the flow of a noncompressible fluid through a pipe of constant diameter, the total mechanical-energy balance can be reduced to the following form ... 

Consider a fluid with bulk temperature T, flowing in a cylindrical tube of diameter D, with constant wall temperature 7 ,. An energy balance on a short section of the tube yields... 

The slope of the mean fluid temperature T , on a T-x diagram can be determined by applying the steady-flow energy balance to a tube slice of thickness dx shown in Fig. 8-12. It gives... 

Consider the heating of a fluid in a tube of constant cross section whose inner surface is maintained at a constant temperature of T,. We know that the mean temperature of the fluid increases in the flow direction as a result of heal transfer. Tlie energy balance on a differential control volume shown in Fig, 8-12 gives... 

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)... 

Duct Flow of Compressible Fluids Thermodynamics provides equations interrelating pressure changes, velocity, duct cross-sectional area, enthalpy, entropy, and specific volume within a flowing stream. Considered ere is the adiabatic, steady-state, one-dimensional flow of a compressible fluid in the absence of shaft work and changes in potential energy. The appropriate energy balance is Eq. (4-155). With Q, Wj, and Az all set equal to zero,... 

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