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CD Ch08 VariableHeatCapacities

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   Chap. Professional Reference Shelf  549PROFESSIONAL REFERENCE SHELF   R8.3Heat Capacities  AConstant or Mean Heat Capacities   Heat capacities are typically expressed as a function of temperature by thequadratic(R8.3-1)We will now consider both the case of constant (or mean) heat capacitiesand variable heat capacities.For the case of constant or mean heat capacities(R8.3-2)The circumflex denotes that the heat capacities are evaluated at some meantemperature value between T    R   and T    .(R8.3-3)In a similar fashion, we can write the integral involving   i   and in Equa-tion (R8.3-1) as is the mean heat capacity of species i   between T   i  0   and T    :(R8.3-4)Substituting the mean heat capacities into Equation (R8.3-1), the steady-stateenergy balance becomes(8.3-5)In almost all of the systems we will study, the reactants will be entering thesystem at the same temperature; therefore, T   i  0      T   0   . C   P  i  i  i T   i T  2   H  Rx T  ()   H   Rx T   R ()  C  ˆ  P  TT   R  ()  C  ˆ  p  C   p   T d  T   R T   TT   R  --------------------------  T   R    →  T     C    ˆ  P  C   P  i   i   C   P  i   T d  T   i  0  T        i C  ˜   P  i TT  i 0  ()  C  ˜  P  i C  ˜  P  i C   P  i   T d  T  i 0 T   TT  i 0  ------------------------  T  i 0    →  T C    ˜    P   i W  ˙  s  F  A0     i C  ˜   P  i TT  i 0  ()     F  A0  X    H   Rx T   R ()  C  ˆ   P  TT   R  ()  []     Energy balance interms of mean orconstant heatcapacities   550  Chap. BVariable Heat Capacities   We next want to arrive at a form of the energy balance for the case where heatcapacities are strong functions of temperature over a wide temperature range.Under these conditions, the mean values used in Equation (8-30) may not beadequate for the relationship between conversion and temperature. CombiningEquation (8-23) with the quadratic form of the heat capacity, Equation(R8.3-1),(R8.3-1)we find thatIntegrating gives us   (R8.3-7)   whereIn a similar fashion, we can evaluate the heat capacity term in Equation(R8.3-1):(R8.3-8)Substituting Equations (R8.3-7) and (R8.3-8) into Equation (R8.3-5), the formof the energy balance is  (R8.3-9   ) C   P  i  i  i T   i T  2   H  Rx T  ()   H   Rx T   R ()  T   T  2  ()  T d  T    R  T        H  Rx T  ()   H   Rx T   R ()  TT    R ()  2-------  T  2   T   R 2    ()    3-------  T  3   T   R 3    ()  Heat capacity as afunction of temperature  d a ---  D ca ---  C ba ---  B  A  d a ---  D ca ---  C ba ---  B  A  d a ---  D ca ---  C ba ---  B  A  ii 1  n    C   P  i   T d  T   0  T        i  i   i  i T    i  i T  2      ()   T d  T   0  T        i  i TT  0  ()   i  i 2-----------------  T  2   T  02    ()   i  i 3-----------------  T  3   T  03    ()  W  ˙  s  F  A0     i  i TT  0  ()   i  i 2-----------------  T  2   T  02    ()   i  i 3-----------------  T  3   T  03    (       F   A0    X      H   Rx T   R ()  TT    R ()    2-------  T  2   T   R 2    ()    3-------  T  3   T   R 3    ()    Energy balance forthe case of highlytemperature-sensitive heatcapacities   Sec. R8.3Heat Capacities  551    Example R8.3–1.1Production of Acetic Anhydride   Jeffreys,   *   in a treatment of the design of an acetic anhydride manufacturing facility,states that one of the key steps is the vapor-phase cracking of acetone to ketene andmethane:He states further that this reaction is first-order with respect to acetone and that thespecific reaction rate can be expressed by (ER8.3-1.1)where k    is in reciprocal seconds and T    is in kelvin. In this design it is desired to feed8000 kg of acetone per hour to a tubular reactor. The reactor consists of a bank of 1000 1-inch schedule 40 tubes. We will consider two cases:1.The reactor is operated adiabatically   .2.The reactor is surrounded by a heat exchanger    where the heat-transfer coeffi-cient is 110 J/m   2      s      K, and the ambient temperature is 1150 K.The inlet temperature and pressure are the same for both cases at 1035 K and 162 kPa(1.6 atm), respectively. Plot the conversion and temperature along the length of the reactor.   Solution   Let , , and . Rewriting the reaction symboli-cally gives us1.   Mole balance   :2.   Rate law   :(ER8.3-1.2)(ER8.3-1.3)3.   Stoichiometry   (gas-phase reaction with no pressure drop):(ER8.3-1.4)4.   Combining   yields(ER8.3-1.5)(ER8.3-1.6)   *   G. V. Jeffreys,  A Problem in Chemical Engineering Design: The Manufacture of Ace-tic Anhyride   , 2nd ed. (London: Institution of Chemical Engineers, 1964). CH 3 COCH 3 CH 2 COCH 4  → k  ln34.3434222 , T  -----------------  ACH 3 COCH 3  BCH 2 CO  CCH 4  ABC  → dX dV  ------ r  A  F  A 0    r  A  kC  A  A C  A0 1  X   () T  0 1     X   () T  ---------------------------------     y A0  1111  ()  r  A  kC  A0 1  X   () 1  X   ------------------------------  T    0   T    -----    dX dV  ------ r  A   F  A0 ---------- k  v 0 ----- 1    X       1    X     -------------      T    0   T    -----      552  Chap. To solve the differential equation (ER8.3-1.6), it is first necessary to use theenergy balance to determine T    as a function of  X    . 5.   Energy balance:   CASE I.ADIABATIC OPERATION   For no work done on the system, , and adiabatic operation, (i.e.,), Equation (8-56) becomes   (   ER8.3   -1.7)   where in generalFor acetone decompositionEquivalent expressions exist for and .Because only A enters,and Equation (ER8.3-1.7) becomesIntegrating gives   (   ER8.3   -1.8)   6.   Calculation of mole balance parameters:   W   s 0  Q 0  U  0  dT dV  ------ r  A  ()    H   Rx T   R ()  T   T  2    () T d  T   R T           F    A0      i   C     P  i    X       C     P       ()   ---------------------------------------------------------------------------------------------------------------------------------    d a ---  D ca ---  C ba ---  B  A  B  C  A  i C   P  i C   P  A  dT dV  ------ r  A  ()    H   Rx T   R ()  T   T  2    () T d  T   R T           F    A0   C     P    A    X     C   P     () ---------------------------------------------------------------------------------------------------------------------------------    dT dV  ------ r  A  ()    H   Rx  TT   R  ()    2-------  T  2   T   R 2    ()    3-------  T  3   T   R 3    ()        F    A0      A      A   T       A   T    2    X     C   P     () ---------------------------------------------------------------------------------------------------------------------------------------------------------------    Adiabatic PFRwith variable heatcapacities  F  A0 8000  kg/h 58  g/mol -------------------------137.9  kmol/h 38.3  mol/s  C  A0  P  A0  RT  --------162  kPa 8.31  kPam   3      kmolK     -------------------- 1035  K  () ---------------------------------------------------------0.0188  kmolm   3   ------------18.8  mol/m 3    v   0    F    A0   C    A0   ---------2.037   m   3   /s   
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