barcode generator vb.net code Applications of Thermodynamics to Flow Processes in Software

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CHAPTER 7 Applications of Thermodynamics to Flow Processes
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Flow processes inevitably result from pressure gradients within the fluid Moreover, temperature, velocity, and even concentration gradients may exist within the flowing fluid This contrasts with the uniform conditions that prevail at equilibrium in closed systems The distribution of conditions in flow systems requires that properties be attributed to point masses of fluid Thus we assume that intensive properties, such as density, specific enthalpy, specific entropy, etc, at a point are determined solely by the temperature, pressure, and composition at the point, uninfluenced by gradients that may exist at the point Moreover, we assume that the fluid exhibits the same set of intensive properties at the point as though it existed at equilibrium at the same temperature, pressure, and composition The implication is that an equation of state applies locally and instantaneously at any point in a fluid system, and that one may invoke a concept of local state, independent of the concept of equilibrium Experience shows that this leads for practical purposes to results in accord with observation The equations of balance for open systems from Chaps 2 and 5 are summarized here in Table 71 for easy reference Included are Eqs (71) and (72), restricted forms of the mass balance These equations are the basis for the thermodynamic analysis ofprocesses in this and the next two chapters When combined with thermodynamic property statements, they allow calculation of process rates and system states
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71 DUCT FLOW OF COMPRESSIBLE FLUIDS
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Such problems as the sizing of pipes and the shaping of nozzles require application of the momentum principle of fluid mechanic^,^ and therefore do not lie within the province of thermodynamics However, thermodynamics does provide equations that interrelate the changes occurring in pressure, velocity, cross-sectional area, enthalpy, entropy, and specific volume of a flowing stream We consider here the adiabatic, steady-state, one-dimensional flow of a compressible fluid in the absence of shaft work and of changes in potential energy The pertinent thermodynamic equations are first derived; they are then applied to flow in pipes and nozzles The appropriate energy balance is Eq (232) With Q, W, and Az all set equal to zero,
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In differential form,
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The continuity equation, Eq (227), is also applicable Since m is constant, its differential form is:
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'see W L McCabe, J C Smith, and P Harriott, Unit Operations o Chemical Engineering, 5th ed, Sec 2, f McGraw-Hill, New York, 1993; R H Peny and D Green, Perry's Chemical Engineers' Handbook, 7th ed, Sec 6, McGraw-Hill, New York, 1997
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Table 71 Equations of Balance
Balance Equations Balance Equations for General Equations of Balance Steady-Flow Processes for Single-Stream Steady-Flow Processes
d(mS),, dt
Q, + A ( S ~ + - ) ~ ~ 2 = SGL 0 L
Toj
23 8
CHAPTER 7 Applications o Thermodynamics to Flow Processes f The fundamental property relation appropriate to this application is:
(68) In addition, the specific volume of the fluid may be considered a function of its entropy and pressure: V = V(S, P) Then,
dH= TdS+VdP
This equation is put into more convenient form through the mathematical identity:
Substituting for the two partial derivatives on the right by Eqs (32) and (617) gives:
where p is the volume expansivity The equation derived in physics for the speed of sound c in a fluid is:
Substituting for the two partial derivatives in the equation for d V now yields:
Equations (73),(74),(68),and (75) relate the six differentials-dH, du, dV, dA, dS, and d P With but four equations, we treat d S and dA as independent, and develop equations that express the remaining differentials as functions of these two First, Eqs (73) and (68) are combined: T d S f V d P = -udu
(76)
Eliminating d V and du from Eq (74) by Eqs (75) and (76) gives upon rearrangement:
where M is the Mach number, defined as the ratio of the speed of the fluid in the duct to the speed of sound in the fluid, ulc Equation (77) relates d P to d S and dA Equations (76) and (77) are combined to eliminate V d P :
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