Energy balance for a closed system

0 = rate of energy transfer accompanying mass transfer in = rate of mass transfer across system boundaries Q = rate of heat transfer Er =rate of energy generation v = fluid velocity through the system W = rate of work done by the system 0 = time 

The relation given by Equation 3.2 has units of energy per unit time. If we divide both sides of this expression by A8 and take the limit as Ad 0, we obtain the integrated form, where units are those of energy  

The quantities designated in Equation 3.5 without the tilde ( ) are integrated values. The bar (-) indicates per unit mass. 

Additional cases applied to the general energy balance are  

Adiabatic. This means no heat exchange (i.e., Q = 0). Examples include insulated systems, small Q in relation to other terms in the energy equation, and very fast processes where there is insufficient time for heat transfer to take place. 

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Write down the energy balance for a closed system in symbols [Eq. (4.23)1, and apply it to solve energy balance problems. 

Figure 4.12 Terms in the energy balance for a closed system.

In deriving the integral mass balance for a closed system in Section 4.2c we eliminated the input and output terms, since by definition no mass crosses the boundaries of a closed system. It is possible, however, for energy to be transferred across the boundaries as heat or work, so that the right side of Equation 7.3-1 may not be eliminated automatically. As with mass balances, however, the accumulation term equals the final value of the balanced quantity (in this case, the system energy) minus the initial value of this quantity. Equation 7,3-1 may therefore be written... 

The starting points for the analysis are the energy and entropy balances for a closed system ... 

For the process of converting an ideal gas into a real substance, show that the heat and work effects presented in 6.3.2 and 6.3.3 are consistent with the energy balance on a closed system. 

As a statement of energy conservation, the first law is the starting point for all energy balances in a closed system. Problems of this type typically require the calculation of heat and work. The amount of heat that is exchanged can, under special conditions, be related to internal energy or enthalpy ... 

A certain amount of heat dQ) must be supplied to or removed from the closed adsorption system of Fig. 1 to maintain system isothermality when the GSE of component i is changed by a differential amount (dni ). By applying the conventional energy balance for the closed system [II], one gets... 

The energy balance for an open system contains all the terms associated with an ener balance for a closed system, but we must also account for the energy change in the system associated with the streams flowing into and out of the system. To accomplish this task, we consider the case of the generic open system illustrated in Figure 2.9. This open system happens to have two streams in and two streams out however, the balances developed here will be true for any number of inlet or outlet streams. 

For the energy balance in a flow system we therefore assume that we can make an enthalpy balance on the contents of the reactor. We can write the rate of enthalpy generation Aff in any flowing or closed system as... 

The first law of thermodynamics provides a description of the energy balance for a given process the second law provides a criterion for deciding whether or not the process will occur spontaneously. The second law of thermodynamics defines the entropy change (A5, in units of J K l) associated with a change in a closed system in terms of the heat absorbed by the system at constant temperature T ... 

An integral energy balance may be derived for a closed system between two instants of time. Since energy can neither be created nor destroyed, the generation and consumption terms of the general balance (4.2-1) drop out, leaving... 

The first law of thermodynamics for a closed system (which we will generally refer to as the energy balance) between two instants of time is... 

If a system expands in volume by an amount AV(m- ) against a constant restraining pressure P(N/m ), a quantity PAV (J) of energy is transferred as expansion work from the system to its surroundings. Suppose that the following four conditions are satisfied for a closed system (a) the system expands against a constant pressure (so that Ap = 0) (b) A k == 0 (c) A p = 0 and (d) the only work done by or on the system is expansion work. Prove that under these conditions, the energy balance simplifies ioQ = A//. 

The energy balance for a homogeneous closed system of n moles is ... 

This relation is recognized from introductory subjects on thermodynamics. Recall that in equilibrium thermodynamics a local formulation is generally not needed, since the intensive state variables are independent of the space coordinates. This fundamental formulation of the total energy balance is known as the first law of thermodynamics for a closed system, which expresses the fundamental physical principle that the total energy of the system, Etotab is conserved (a postulate). 

We begin with the application of the first law of thermodynamics first to a dosed system and then to an open system. A system is any bounded portion of the universe, moving or stationary, which is chosen for the application of the various thermodynamic equations. For a closed system, in which no mass crosses the system boundaries, the change in total energy of the system, dE, is equal to the heat flow to the system. 8Q. minus the work done by the system on the surroundings. W. For a closed sy.sreni. the energy balance is... 

Although few processes are truly reversible, it is sometimes useful to model them to be so. When this is done, it is clear that any computations made based on Eqs. 4.1-5 or 4.1-9, with 5oen = en = 0, will only be approximate. However, these approximate results may be very useful, since the term neglected (the entropy generation) is of known sign, so we will know whether our estimate for the heat, work, or any state variable is an upper or lower bound to the true value. To see this, consider the energy and entropy balances for a closed, isothermal, constant-volume system ... 

It is also possible to develop the equilibrium and stability conditions for systems subject to other constraints. For a closed system at constant temperature and volume, the energy and entropy balances are... 

The contribution to the energy from the surface is usually written as a A, where a is the surface tension for the liquid-vapor interface (or the interfacial tension for a liquid-liquid interface) and A is the surface area. The a A contribution is the two-dimensional analogue of the P V term For bulk fluids. Including the effect of changing surface area in the energy balance, just as we have included the effect of changing volume, gives for a closed system without shaft work. ... 

Work out the energy balance for a throttle valve (Eq, 4.39), using the closed-system form of the first law. Choose as your system 1 kg of material flowing down the line. 

This is precisely the form of the first law for a closed system of constant volume that exchanges heat and shaft work with the surroundings (see eg. r. oll The reason that the PFwork does not appear here is that the form of the energy balance in eg. (6.i7l implicitly assumes that the system has constant volume, i.e., its boundaries are rigid and thus the system is prevented from exchanging any PF work. For an open system with movable boundaries (a rubber balloon, for example), eg. (6.17) must be amended to include PFwork in addition to any shaft work. 

While the energy balance (or first law) is required to evaluate the energy changes of the system, the ideal gas law describes the relationship between the gas properties. For a closed system, the energy balance was written as... 

The first law of thermodynamics states that the total energy in the universe is a constant. Energy balances have been developed for closed systems and for open systems. For example, the integral equation of the first law for a closed system, written in extensive form, is ... 

In open systems, the mass that crosses the boundary between the surroundings and the system always contributes to two terms in the energy balance internal energy and flow (Pc) work. Since these terms are always coupled, it is convenient to define a property that includes both terms. In this way we never need to explicitly account for flow work. Likewise, enthalpy is a convenient property for a closed system undergoing a process at constant pressure. In this case, we need to consider both the change in internal energy and the Pv work. 

Let s consider again the calculation of internal energy. This time we begin with the fundamental postulates of thermodynamics. For a closed system undergoing a reversible process with only Pv work, the relations developed in Sections 2.3 and 3.3 can be applied to the differential energy balance, Equation (2.14). Hence, the first law and second laws are combined to give ... 

The general criterion of chemical reaction equiUbria is the same as that for phase equiUbria, namely that the total Gibbs energy of a closed system be a minimum at constant, uniform T and P (eq. 212). If the T and P of a siagle-phase, chemically reactive system are constant, then the quantities capable of change are the mole numbers, n. The iadependentiy variable quantities are just the r reaction coordinates, and thus the equiUbrium state is characterized by the rnecessary derivative conditions (and subject to the material balance constraints of equation 235) where j = 1,11,.. ., r ... 

Define the terras closed process system, open process system, isothermal process, and adiabatic process. Write the first law of thermodynamics (the energy balance equation) for a closed process system and state the conditions under which each of the five terms in the balance can be neglected. Given a description of a closed process system, simplify the energy balance and solve it for whichever term is not specified in the process description. 

At this point, you can perform energy balance calculations only for systems in which A / (closed system) or AH (open system) can be neglected and for nonreactive systems involving species for which tables of f/ or H are available. Energy balance procedures for other types of systems are presented in Chapters 8 and 9. 

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