LESSON 10:
The continuity equation, (1-41), is also applicable. Since m. is

Thus the pressure decreases and the velocity increases in the

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Lecture 10


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Lecture 10

  1. 1. THERMODYNAMICS OF BIOLOGICAL SYSTEMS LESSON 10: CALCULATION OF FLOW PROCESSES BASED ON ACTUAL PROPERTY CHANGES In our earlier lessons we have learnt that the thermodynamics At this point, you should be able to visualize a pipe to the fluid of flow is based on mass, energy and entropy balances. The flow. Try to question what parameters will allow a smooth flow, application of these balances to specific processes will be what factors determine the rate of flow and how are the energy considered in this lesson. The discipline underlying the study of parameters affected. These questions lead us to the fact that any flow is fluid mechanics, which encompasses not only the modification in the pipes should bring about a change in the balances of thermodynamics but also the linear-momentum flow process. Can you now visualize any biological system that principle (Newton’s second Law). This makes fluid mechanics a is subjected to flow processes? What if there are modifications broader field of study. in such systems? How is flow then affected? The distinction between thermodynamics problems and fluid mechanics problems depends on whether this principle is required for solution. Those problems whose solutions depend only on mass conservation and on the laws of thermodynamics are commonly set apart from the study of fluid mechanics and are treated in courses in thermodynamics. Fluid mechanics then deals with the broad spectrum of problems which require application of the momentum principle. This division is arbitrary, but it is traditional and convenient. Consider for example the flow of gas through a pipeline. If the states and thermodynamic properties of the gas entering and leaving the pipe line are known, then application of the first law establishes the magnitude of the energy exchange with the surroundings of the pipeline. The mechanism of the process, the details of flow, and the path actually followed by the fluid between entrance and exit are not pertinent to this calculation. On the other hand, if one has only incomplete knowledge of the initial or final state of the gas, then more detailed informa- tion about the process is needed before any calculations are made. For example, the exit pressure of the gas may not be vsd/norm.html specified. In this case, one must apply the momentum principle of fluid mechanics, and this requires an empirical or theoretical Duct Flow of Compressible Fluids expression for the shear stress at the pipe wall. Problems such as sizing of pipes and shaping of the nozzles requires Flow processes inevitably result from pressure gradients within application of the momentum principle of fluid mechanics. This does the fluid. Moreover, temperature, velocity, and even concentra- not lie within the subject of thermodynamics. However, thermody- tion gradients may exist within the flowing fluid. This contrasts namics does provide equations that interrelate the changes with the uniform conditions that prevail at equilibrium in occurring in pressure, velocity, cross-sectional area, closed systems. The distribution of conditions in flow systems enthalpy, entropy and specific volume of flowing stream. requires that properties be attributed to point masses of fluid. We consider here the adiabatic, steady-state, one-dimen- Thus we assume that intensive properties, such as density, sional flow of compressible fluid in the absence of shaft specific enthalpy, specific entropy, etc., at a point are determined work and changes in potential energy. The pertinent solely by the temperature, pressure and composition at the thermodynamic equations are first derived; they are then applied point, uninfluenced by gradients that may exist at the point. to pipes and nozzles. Moreover, we assume that the fluid exhibits the same set of The appropriate energy balance is equation (1-46). With Q, Ws intensive properties at the point as though it existed at equilib- and z all set to zero, rium at the same temperature, pressure and composition. The implication is that an equation of state applies locally and ∆u 2 instantaneously at any point in a fluid system and that one may ∆H + =0 invoke a concept of local state, independent of the concept of 2 equilibrium. Experiences shows that this leads for practical In differential form, purposes to results in accord with observation. dH = -udu (2-57) © Copy Right: Rai University 2.202 39
  2. 2. The continuity equation, (1-41), is also applicable. Since m. is THERMODYNAMICS OF BIOLOGICAL SYSTEMS (1 − M )VdP + 1 + βu  2 constant, its differential form is: u2 2   TdS −  dA = 0 (2-61) d(uA/V) = 0 C P  A dV du dA where M is the Mach number, defined as the ratio of the Or − − =0 (2-58) fluid in the duct to the speed of sound in the fluid, u/c. V u A Equation (2-61) relates dP to dS and dA. The fundamental property relation appropriate to this applica- Equation 2-60 and 2-61 are combined to eliminate VdP: tion is: dH = TdS + VdP  βu 2  In addition, the specific volume of the fluid may be considered  +M2   CP   1 u 2 a function of its entropy and pressure: udu −  TdS +   dA = 0 1− M 2 1− M  A 2 (2-62) V = V (s,p).        ∂V   ∂V  Then, dV =   dS +   dP This equation relates to dS and dA. Combined with eq 2-57 it  ∂S  P  ∂P  S relates dH to dS and dA, and combined with 2-58 it relates dV This equation is put into more convenient form through the to these same independent variables. mathematical identity: The differentials in the preceding equations represent changes in the fluid as it traverses a differential length of its path. If this  ∂V   ∂V   ∂T  length is dx, then each of the equations of flow may be divided   =     ∂S  P  ∂T  P  ∂S  P through by dx. Equations 7.7 and 7.8 then become: ( ) dP + T 1 + βu  dS u 2 dA 2 Substituting for the two partial derivatives on the right by eqs. V 1− M 2     dx − A dx = 0 (2-63)  ∂S  C dx  C P  2.3 and gives:   = P  ∂T  P T  βu 2   +M2    dS  1  u dA 2  ∂V  βVT du −T CP   = u dx 1− M 2  dx +  1 − M 2    A dx =0 (2-64) where is the volume expansivity  ∂S  P Cp       According to the second law, the irreversibilities due to fluid The equation derived in physics for the speed of sound c in a friction in adiabatic flow cause an entropy increase in the fluid in fluid is: the direction of flow. In the limit as the flow approaches reversibility, this increase approaches zero. In general, then,  ∂P   ∂V  V 2  c 2 = −V 2   or   = − 2  c  dS  ∂V  S  ∂P  S   ≥0 dx substituting for the two partial derivatives in the equation for Pipe Flow dV now yields: For the case of steady-state adiabatic flow in a horizontal pipe of constant cross-sectional area, dA/dx = 0, and eqn.. dV βT V 2-63 and 2-64 reduces to: = dS − 2 dP (2-59) V CP c  βu 2   βu 2  Equations 2-57, 2-58, 2-48 and 2-59 relate the six differentials- 1+   +M 2  dH, du, dv, dA, dS, and dP. With but four equations, we treat dP T CP  dS du  CP  dS =−   dx u dx = T  1 − M 2  dx dS and dA as independent, and develop equations that express dx V 1− M 2     the remaining differentials as functions of these two. First, eqns     2-57 and 2-48 are combined:     TdS + VdP = -udu (2-60) For subsonic flow, M2 < 1, and all quantities on the right sides Eliminating dV and du from eqs. 2-58 by eqs. 2-59 and 2-60 dP gives upon rearrangement: of these equations are positive whence, dx <0 du and >0 dx © Copy Right: Rai University 40 2.202
  3. 3. Thus the pressure decreases and the velocity increases in the THERMODYNAMICS OF BIOLOGICAL SYSTEMS direction of flow. However, the velocity cannot increase indefinitely. If the velocity were to exceed the sonic value, then the above inequalities would reverse. Such a transition is not possible in a pipe of constant cross-sectional area. For subsonic flow, the maximum fluid velocity obtainable in a pipe of constant cross section is the speed of sound, and this value is reached at the exit of the pipe. At this point dS/dx reaches its limiting value of zero. Given a discharge pressure low enough for the flow to become sonic, lengthening the pipe does not alter this result; the mass rate of flow decreases so that the sonic velocity is still obtained at the outlet of the lengthened pipe. The equations for pipe flow indicate that when flow is super- sonic the pressure increases and the velocity decreases in the direction of flow. However, such a flow regime is unstable, and when a supersonic stream enters a pipe of constant cross section, a compression shock occurs, the result of which is an abrupt and finite increase in pressure and decrease in velocity to a subsonic value. The limitations observed for flow in pipes do not extend to properly designed nozzles, which bring about the interchange of internal and kinetic energy of a fluid as the result of a changing cross-sectional area available for flow. The relation between nozzle length and cross-sectional area is not susceptible to thermodynamics analysis, but is a problem in fluid mechan- ics. In a properly designed nozzle the area changes with length in such a way as to make the flow nearly frictionless. Problems 1. Consider the steady-state, adiabatic, irreversible flow flow of an incompressible liquid in a horizontal pipe of constant cross-sectional area. Show that: a. The velocity is constant b. The temperature increases in the direction of flow c. The pressure decreases in the direction of flow References 1. J. M. Smith, H. C. Van Ness, M. M. Abbott, Adapted by B. I. Bhatt, Introduction To Chemical Engineering Thermodynamics, Sixth Edition, Tata McGraw-Hill Publishing Company Ltd, New Delhi Notes © Copy Right: Rai University 2.202 41