Flow term energy balance

Two approaches to this equation have been employed. (/) The scalar product is formed between the differential vector equation of motion and the vector velocity and the resulting equation is integrated (1). This is the most rigorous approach and for laminar flow yields an expHcit equation for AF in terms of the velocity gradients within the system. (2) The overall energy balance is manipulated by asserting that the local irreversible dissipation of energy is measured by the difference ... 

The first term on the right-hand side of Eq. (14.113) comes from the inertial forces. Because of the pressure drop the density of gas decreases in the di rection of the flow and therefore, on the basis of mass balance of gas flow, the velocity v increases along the flow. If the pipe is isolated, then the flow can be treated as adiabatic, which on the basis of energy balance implies that along the flow we have... 

In considering the flow in a pipe, the differential form of the general energy balance equation 2.54 are used, and the friction term 8F will be written in terms of the energy dissipated per unit mass of fluid for flow through a length d/ of pipe. In the first instance, isothermal flow of an ideal gas is considered and the flowrate is expressed as a function of upstream and downstream pressures. Non-isothermal and adiabatic flow are discussed later. 

Methods have been given for the calculation of the pressure drop for the flow of an incompressible fluid and for a compressible fluid which behaves as an ideal gas. If the fluid is compressible and deviations from the ideal gas law are appreciable, one of the approximate equations of state, such as van der Waals equation, may be used in place of the law PV = nRT to give the relation between temperature, pressure, and volume. Alternatively, if the enthalpy of the gas is known over a range of temperature and pressure, the energy balance, equation 2.56, which involves a term representing the change in the enthalpy, may be employed ... 

With only component B in the feed, the flow term in the energy balance becomes... 

This term includes, power, steam, compressed air, cooling and process water, and effluent treatment unless costed separately. The quantities required can be obtained from the energy balances and the flow-sheets. The prices should be taken from Company records, if available. They will depend on the primary energy sources and the plant location. The figures given in Table 6.5 can be used to make preliminary estimates. The current cost of utilities supplied by the utility companies electricity, gas and water, can be obtained from their local area offices. 

There are a variety of limiting forms of equation 8.0.3 that are appropriate for use with different types of reactors and different modes of operation. For stirred tanks the reactor contents are uniform in temperature and composition throughout, and it is possible to write the energy balance over the entire reactor. In the case of a batch reactor, only the first two terms need be retained. For continuous flow systems operating at steady state, the accumulation term disappears. For adiabatic operation in the absence of shaft work effects the energy transfer term is omitted. For the case of semibatch operation it may be necessary to retain all four terms. For tubular flow reactors neither the composition nor the temperature need be independent of position, and the energy balance must be written on a differential element of reactor volume. The resultant differential equation must then be solved in conjunction with the differential equation describing the material balance on the differential element. 

The input and output terms of equation 1.5-1 may each have more than one contribution. The input of a species may be by convective (bulk) flow, by diffusion of some kind across the entry point(s), and by formation by chemical reaction(s) within the control volume. The output of a species may include consumption by reaction(s) within the control volume. There are also corresponding terms in the energy balance (e.g., generation or consumption of enthalpy by reaction), and in addition there is heat transfer (2), which does not involve material flow. The accumulation term on the right side of equation 1.5-1 is the net result of the inputs and outputs for steady-state operation, it is zero, and for unsteady-state operation, it is nonzero. 

Taken together, applied to a fluid flowing inside the tube, these terms lead to the energy balance for steady-state operation in the form ... 

Inputs + Sources = Outputs + Sinks + Accumulations where each of these terms may be a quantity or a rate. Inputs and Outputs are accomplished by crossing the boundary of the reference volume. In case of mass transfer this occurs by bulk flow and diffusion. Sources and Sinks are accretions and depletions of a species without crossing the boundaries. In a mass and energy balance, sinks are the rate of reaction, rdVr, or a rate of enthalpy change, AHrpdC. Accumulation is the time derivative of the content of the species within the reference volume, for example, (<9C/3t)dVr or... 

Orthogonal factorizations may be applied to resolve problem (5.3) if the system of equations cp(x, u) = 0 is made up of linear mass balances and bilinear component and energy balances. After replacing the bilinear terms of the original model by the corresponding mass and energy flows, a linear data reconciliation problem results. 

It is found convenient to base compressible flow calculations on an energy balance per unit mass of fluid and to work in terms of the fluid s specific volume V rather than the density p. The specific volume is the volume per unit mass of fluid and is simply the reciprocal of the density ... 

In general, the cell energy balance states that the enthalpy flow of the reactants entering the cell will equal the enthalpy flow of the products leaving the cell plus the sum of three terms (1) the net heat generated by physical and chemical processes within the cell, (2) the dc power output from the cell, and the rate of heat loss from the cell to its surroundings. 

Thus, even though each term in the energy-balance equation is now dimensionless, we associate each term with a term in the original energy-balance equation T — To)/J represents the heat flow out rninus the heat flow in, k T — Tc)I J represents the heat transfer through the walls, and X(T) represents the rate of heat generation by reactiou... 

For R simultaneous reactions we have R different Xj values and R simultaneous maSS-balance equations to solve along with the energy-balance equation. In the energy-balance equation the flow and heat removal terms are identical, but the energy generation has terms for each of the R reactions,... 

This is the equation of energy balance, which states that the internal energy of a fluid changes because of the flow of energy into it, because of the pressure-volume work done on the surroundings by the fluid element, and because of loss of energy by viscous dissipation.6 The energy-balance equation may be written in an alternate form in terms of the temperature ... 

The differential energy balances of Eqs. (6.10) and (6.15) with the friction term of Eq. (6.18) can be integrated for compressible fluid flow under certain restrictions. Three cases of particular importance are of isentropic or isothermal or adiabatic flows. Equations will be developed for them for ideal gases, and the procedure for nonidcal gases also will be indicated. 

As an illustration, we will derive the viscous dissipation terms in the energy balance using a simple shear flow system such as the one shown in Fig. 5.6. 

S based on experiments with water in turbulent flow, in channels icient roughness that there is no Reynolds number effect. The hydraulic radius approach may be used to estimate a friction factor with which to compute friction losses. Under conditions of uniform flow where liquid depth and cross-sectional area do not vary significantly with position in the flow direction, there is a balance between gravitational forces and wall stress, or equivalently between frictional fosses and potential energy change. The mechanical energy balance reduces to tv = g(zx — z2). In terms of the friction factor and hydraulic diameter or hydraulic radius,... 

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