**Introduction to Thermal Systems...echanics, and Heat...**

Livro: Introducao à Engenharia de Sistemas Térmicos (Inglês)

(Parte **1** de 6)

5CONTROL VOLUME ANALYSIS USING ENERGY

Introduction…

The objectiveof this chapter is to develop and illustrate the use of the control volume forms of the conservation of mass and conservation of energy principles. Mass and energy balances for control volumes are introduced in Secs. 5.1 and 5.2, respectively. These balances are applied in Sec. 5.3 to control volumes at steady state.

Although devices such as turbines,pumps,and compressors through which massflows can be analyzed in principle by studying a particular quantity of matter (a closed system) as it passes through the device,it is normally preferable to think of a region of space through which mass flows (a control volume). As in the case of a closed system,energy transfer across the boundary of a control volume can occur by means of work and heat. In addition,another type of energy transfer must be accounted for—the energy accompanying mass as it enters or exits.

Conservation of Mass for a Control Volume

In this section an expression of the conservation of mass principle for control volumes is developed and illustrated. As a part of the presentation,the one-dimensional flow model is introduced.

Developing the Mass Rate Balance

The mass rate balance for control volumes is introduced by reference to Fig. 5.1,which shows a control volume with mass flowing in at iand flowing out at e,respectively. When applied to such a control volume,the conservation of massprinciple states

£time rate of change of mass contained within the control volume at time t

§ £time rate of flow of mass in across inlet i at time t

§ £time rate of flow of mass out across exit e at time t §

5.1 conservation of mass chapter objective

Dashed line defines the control volume boundary

Inlet i

Exit e Figure 5.1One-inlet,one-exit control volume.

Denoting the mass contained within the control volume at time tby mcv(t),this statement of the conservation of mass principle can be expressed in symbols as

where and are the instantaneous mass flow ratesat the inlet and exit,respectively. As for the symbols and the dots in the quantities and denote time rates of transfer. In SI,all terms in Eq. 5.1are expressed in kg/s. Other units employed in this text are lb/s and slug/s.

In general,there may be several locations on the boundary through which mass enters or exits. This can be accounted for by summing,as follows:

Equation 5.2is the mass rate balancefor control volumes with several inlets and exits.

It is a form of the conservation of mass principle commonly employed in engineering. Other forms of the mass rate balance are considered in discussions to follow.

One-dimensional Flow

When a flowing stream of matter entering or exiting a control volume adheres to the following idealizations,the flow is said to be one-dimensional:(1) The flow is normal to the boundary at locations where mass enters or exits the control volume. (2) Allintensive properties,including velocity and specific volume,are uniform with position(bulk average values) over each inlet or exit area through which matter flows. In subsequent control volume analyses in thermodynamics we routinely assume that the boundary of the control volume can be selected so that these idealizations are appropriate. Accordingly,the assumption of one-dimensional flow is not listed explicitly in the accompanying solved examples.

Figure 5.2illustrates the meaning of one-dimensional flow. The area through which mass flows is denoted by A. The symbol V denotes a single value that represents the velocity of the flowing air. Similarly Tand vare single values that represent the temperature and specific volume,respectively,of the flowing air. When the flow is one-dimensional,the mass flow rate can be evaluated using dmcv dt ai em# i dmcv mass rate balance one-dimensional flow

Air compressor

Air i

AirV, T, v Area = A

Figure 5.2Figure illustrating the one-dimensional flow model.

mass flow rates or in terms of density (5.3b)

When area is in m2,velocity is in m/s,and specific volume is in m3/kg,the mass flow rate found from Eq. 5.3ais in kg/s,as can be verified.

The product AV in Eqs. 5.3is the volumetric flow rate.The volumetric flow rate is expressed in units of m3/s or ft3/s.

Steady-state Form

Many engineering systems can be idealized as being at steady state,meaning that allproperties are unchanging in time. For a control volume at steady state,the identity of the matter within the control volume changes continuously,but the total amount present at any instant remains constant,so and Eq. 5.2reduces to

That is,the total incoming and outgoing rates of mass flow are equal.

Equality of total incoming and outgoing rates of mass flow does not necessarily mean that a control volume is at steady state. Although the total amount of mass within the control volume at any instant would be constant,other properties such as temperature and pressure might be varying with time. When a control volume is at steady state,everyproperty is independent of time. Note that the steady-state assumption and the one-dimensional flow assumption are independent idealizations. One does not imply the other.

The following example illustrates an application of the rate form of the mass balance to a control volume at steady state. The control volume has two inlets and one exit.

volumetric flow rate steady state with a mass flow rate of 40 kg/s. At inlet 2,liquid water at p2 7 bar,T2 40 C enters through an area A2 25 cm2. Saturated liquid at 7 bar exits at 3 with a volumetric flow rate of 0.06 m3/s. Determine the mass flow rates at inlet 2 and at the exit,in kg/s,and the velocity at inlet 2,in m/s.

Solution Known:A stream of water vapor mixes with a liquid water stream to produce a saturated liquid stream at the exit. The states at inlets and exit are specified. Mass flow rate and volumetric flow rate data are given at one inlet and at the exit,respectively.

Find:Determine the mass flow rates at inlet 2 and at the exit,and the velocity V2.

Schematic and Given Data:

Assumption:The control volume shown on the accompanying figure is at steady state.

Example 5.1Feedwater Heater at Steady State

3 Control volume boundary

Saturated liquid

(AV)3 = 0.06 m3/s Figure E5.1

Example 5.2Filling a Barrel with Water

Water flows into the top of an open barrel at a constant mass flow rate of 30 lb/s. Water exits through a pipe near the base with a mass flow rate proportional to the height of liquid inside:where Lis the instantaneous liquid height,in ft. The area of the base in 3 ft2,and the density of water is 62.4 lb/ft3. If the barrel is initially empty,plot the variation of liquid height with time and comment on the result.

Solution (CD-ROM)

Analysis: The principal relations to be employed are the mass rate balance (Eq. 5.2) and the expression (Eq. 5.3a). At steady state the mass rate balance becomes

Solving for The mass flow rate is given. The mass flow rate at the exit can be evaluated from the given volumetric flow rate

The mass flow rate at inlet 2 is then

For one-dimensional flow at 2,so

State 2 is a compressed liquid. The specific volume at this state can be approximated by (Eq. 4.1). From TableT-2 at 40 C,So

At steady state the mass flow rate at the exit equals the sum of the mass flow rates at the inlets. It is left as an exercise t show that the volumetric flow rate at the exit does not equal the sum of the volumetric flow rates at the inlets.

dmcv 0

Example 5.2 illustrates an unsteady,or transient,application of the mass rate balance. In this case,a barrel is filled with water.

Conservation of Energy for a Control Volume

In this section an expression of the conservation of energy principle for control volumes is developed and illustrated.

Dashed line defines the control volume boundary

Inlet i meControl volume zezi

Energy transfers can occur by heat and work ui +Vi 2 ue +Ve 2

Exit e

Figure 5.3Figure used to develop Eq. 5.5.

5.2.1Developing the Energy Rate Balance for a Control Volume

The conservation of energy principle applied to a control volume states:The time rate of change of energy stored within the control volume equals the difference between the total incoming and total outgoing rates of energy transfer.

From our discussion of energy in Chap. 3 we know that energy can enter and exit a closed system by work and heat transfer. The same is true of a control volume. For a control volume, energy also enters and exits with flowing streams of matter. Accordingly,for the one-inlet oneexit control volume with one-dimensional flow shown in Fig. 5.3the energy rate balance is where Ecvdenotes the energy of the control volume at time t. The terms and account, respectively,for the net rate of energy transfer by heat and work across the boundary of the control volume at t. The underlined terms account for the rates of transfer of internal,kinetic,and potential energy of the entering and exiting streams. If there is no mass flow in or out,the respective mass flow rates vanish and the underlined terms of Eq. 5.5drop out. The equation then reduces to the rate form of the energy balance for closed systems:Eq. 3.13.

Evaluating Work for a Control Volume

Next,we will place Eq. 5.5in an alternative form that is more convenient for subsequent applications. This will be accomplished primarily by recasting the work term which represents the net rate of energy transfer by work across allportions of the boundary of the control volume.

Because work is always done on or by a control volume where matter flows across the boundary,it is convenient to separate the work term into two contributions. One is the work associated with the fluid pressure as mass is introduced at inlets and removed at exits. The other contribution,denoted by includes all otherwork effects,such as those associated with rotating shafts,displacement of the boundary,and electrical effects.

Consider the work at an exit eassociated with the pressure of the flowing matter. Recall from Eq. 3.4 that the rate of energy transfer by work can be expressed as the product of a force and the velocity at the point of application of the force. Accordingly,the rate at which work is done at the exit by the normal force (normal to the exit area in the direction of flow) due to pressure is the product of the normal force,peAe,and the fluid velocity,Ve. That is

£time rate of energy transfer by work from the control dEcv

2 gzeb where peis the pressure,Aeis the area,and Veis the velocity at exit e,respectively. A similar expression can be written for the rate of energy transfer by work into the control volume

With these considerations, the work term of the energy rate equation, Eq. 5.5, can be written as

(5.6a) where,in accordance with the sign convention for work,the term at the inlet has a negative sign because energy is transferred into the control volume there. A positive sign precedes the work term at the exit because energy is transferred out of the control volume there. With from Eq. 5.3a,the above expression for work can be written as

(5.6b)

where and are the mass flow rates and viand veare the specific volumes evaluated at the inlet and exit,respectively. In Eq. 5.6b,the terms (pivi) and (peve) account for the work associated with the pressure at the inlet and exit,respectively. The term accounts

for all otherenergy transfers by work across the boundary of the control volume.

The product pvappearing in Eq. 5.6bis commonly referred to as flow workbecause it originates here in a work analysis. However,since pvis a property,the term flow energyalso is appropriate.

5.2.2Forms of the Control Volume Energy Rate Balance

Substituting Eq. 5.6bin Eq. 5.5and collecting all terms referring to the inlet and the exit into separate expressions,the following form of the control volume energy rate balance results:

The subscript “cv”has been added to to emphasize that this is the heat transfer rate over the boundary (control surface) of the control volume.

The last two terms of Eq. 5.7can be rewritten using the specific enthalpy hintroduced in Sec. 4.3.2. With h u pv,the energy rate balance becomes

The appearance of the sum u pvin the control volume energy equation is the principal reason for introducing enthalpy previously. It is brought in solely as a convenience:The algebraic form of the energy rate balance is simplified by the use of enthalpy and,as we have seen,enthalpy is normally tabulated along with other properties.

In practice there may be several locations on the boundary through which mass enters or exits. This can be accounted for by introducing summations as in the mass balance. Accordingly,the energy rate balanceis

Equation 5.9is an accountingbalance for the energy of the control volume. It states that the rate of energy increase or decrease within the control volume equals the difference between the rates of energy transfer in and out across the boundary. The mechanisms of energy transfer are heat and work,as for closed systems,and the energy that accompanies the mass entering and exiting.

dEcv

2 gzeb dEcv

2 gzeb dEcv

2 gzeb em# i flow work flow energy energy rate balance

Equation 5.9provides a starting point for applying the conservation of energy principle to a wide range of problems of engineering importance,including transientcontrol volumes in which the state changes with time. Transient examples include the startup or shutdown of turbines,compressors,and motors. Additional examples are provided by containers being filled or emptied,as considered in Example 5.2 and in the discussion of Fig.2.3. Because property values,work and heat transfer rates,and mass flow rates may vary with time during transient operation,special care must be exercised when applying the mass and energy rate balances. Transient control volume applications are beyond the scope of this introductory presentation of engineering thermodynamics. Only steady-state control volumes are studied,as considered next.

Analyzing Control Volumes at Steady State

In this section steady-state forms of the mass and energy rate balances are developed and applied to a variety of cases of engineering interest. Steady-state cases are commonly encountered in engineering.

5.3.1Steady-state Forms of the Mass and Energy Rate Balances

For a control volume at steady state,the conditions of the mass within the control volume and at the boundary do not vary with time. The mass flow rates and the rates of energy transfer by heat and work are also constant with time. There can be no accumulation of mass within the control volume, so and the mass rate balance, Eq. 5.2, takes the form

Furthermore,at steady state so Eq. 5.9can be written as

(5.10a)

Alternatively (5.10b)

Equation 5.4asserts that at steady state the total rate at which mass enters the control volume equals the total rate at which mass exits. Similarly,Eqs. 5.10assert that the total rate at which energy is transferred into the control volume equals the total rate at which energy is transferred out.

Many important applications involve one-inlet,one-exit control volumes at steady state.

It is instructive to apply the mass and energy rate balances to this special case. The mass rate balance reduces simply to That is,the mass flow must be the same at the exit,2, as it is at the inlet,1. The common mass flow rate is designated simply by Next,applying the energy rate balance and factoring the mass flow rate gives

(Parte **1** de 6)