Page 63 - Bird R.B. Transport phenomena
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48  Chapter 2  Shell Momentum Balances and Velocity  Distributions in Laminar Flow

                           This difficulty  can be overcome if we expand the two exponentials in Taylor series  (see §C2),
                           as  follows:





                                                    8  "'  2!S  '  3!5  •  J\aS
                                               1 :  +     2     3
                                                2
                                             pgd cosp
                                              g8 2  cos
                                             P
                                                                                                (2.2-28)
                           which is in agreement with Eq. 2.2-18.
                               From Eq. 2.2-27 it may be shown that the average velocity  is




                           The  reader may verify  that this result simplifies  to Eq. 2.2-20 when a goes to zero.




      §2.3  FLOW THROUGH A        CIRCULAR     TUBE

                           The  flow  of  fluids  in circular tubes  is encountered frequently  in physics,  chemistry, biol-
                            ogy,  and  engineering.  The laminar  flow  of  fluids  in  circular  tubes  may  be  analyzed  by
                            means  of  the  momentum balance  described  in  §2.1. The  only  new  feature  introduced
                            here is  the use  of  cylindrical  coordinates, which  are the natural coordinates for  describ-
                            ing positions  in a pipe  of circular cross  section.
                               We  consider  then the steady-state, laminar flow  of  a  fluid  of  constant density  p and
                            viscosity  /x in a vertical  tube of length  L and radius  R. The liquid flows downward  under
                            the  influence  of  a pressure  difference  and gravity; the coordinate system  is that shown  in
                            Fig. 2.3-1. We  specify  that the tube  length  be very  large  with  respect  to the tube  radius,
                            so that "end  effects"  will be unimportant throughout most  of  the tube; that is, we  can  ig-
                            nore the fact  that at the tube entrance and exit the flow will not necessarily  be parallel  to
                            the  tube wall.
                               We  postulate that v z  = v (r), v r  = 0, v Q  = 0, and p  = p(z). With  these postulates  it may
                                                   z
                            be  seen  from  Table  B.I  that  the  only  nonvanishing  components  of  т  are  r r2  =  r 2r  =
                            -fiidvjdrl
                               We  select as our system  a cylindrical shell  of thickness Ar and length L and we  begin
                            by listing the various contributions to the z-momentum balance:
                            rate  of z-momentum in                  (2тггАг)(ф )| 2=0            (2.3-1)
                                                                             22
                            across annular surface at z  = 0
                            rate  of z-momentum out                 (2тггДг)(ф )| £              (2.3-2)
                                                                             22 2=
                            across annular  surface  at z  =  L
                            rate  of z-momentum  in             (2ттгЬ){ф )\  =  (2тгг1ф ,)\     (2.3-3)
                                                                       гг  г      г  г
                            across cylindrical surface  at r
                            rate  of z-momentum out        (2тг(г + Ar)L)(0 )|  r+Ar  = (2тгг1ф )\ г+Аг  (2.3-4)
                                                                                      Г2
                                                                         r2
                            across cylindrical surface at r  + Ar
                            gravity  force acting in                  (27rrArL)pg                (2.3-5)
                            z direction on cylindrical shell
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