Page 373 - Bird R.B. Transport phenomena
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§11.5  Dimensional  Analysis  of  the Equations  of  Change  for  Nonisothermal  Systems  355

                 Table  11.5-1  Dimensionless  Groups  in Equations  11.5-7, 8, and  9
                 Special        Forced                    Free        Free
                 cases  —»      convection  Intermediate  convection  convection
                                                          (A)         (B)
                 Choice
                 for  VQ —>                 Щ                         e/'o
                 J±                          1                        Pr
                                Re          Re
                       -  Го)               Gr                            2
                                Neglect                   Gr          GrPr
                                            Re 2
                 [  *   1         1           1

                                RePr        RePr          Pr

                                 Br          Br           Neglect     Neglect
                                RePr        RePr
                        -  To)
                 Notes:
                 "  For forced convection and forced-plus-free  ("intermediate") convection, v  is generally
                                                                      0
                 taken to be the approach velocity  (for flow around submerged  objects) or an average
                 velocity  in the system  (for flow in conduits).
                 b
                  For free convection there are two standard choices for %  labeled as A and  B. In §10.9,
                 Case A arises naturally. Case В proves convenient if the assumption of creeping flow is
                 appropriate, so that Dir/Dt can be neglected  (see Example 11.5-2). Then a new
                 dimensionless pressure difference  Ф = РгФ, different  from & in Eq. 3.7-4, can be
                 introduced, so that when the equation of motion is divided by  Pr, the only dimensionless
                 group appearing in the equation is GrPr. Note that in Case B, no dimensionless groups
                 appear in the equation of  energy.



                    It  is  sometimes  useful  to  think  of  the  dimensionless  groups  as  ratios  of  various
                 forces  or  effects  in the system,  as shown  in Table  11.5-3. For example, the inertial term in
                 the equation  of  motion is p[v  •  Vv]  and  the viscous  term is  /xV v. To get  "typical"  values
                                                                      2
                 of  these terms, replace the variables  by  the characteristic "yardsticks"  used  in construct-
                                                                                       2
                 ing  dimensionless  variables.  Hence  replace  p[v  •  Vv]  by  pvl/l ,  and  replace  /iV v  by
                                                                       0
                 ILV /IO  to  get  rough  orders  of  magnitude. The  ratio  of  these  two  terms  then  gives  the
                   Q
                 Reynolds number, as shown  in the table. The other dimensionless groups  are obtained in
                 similar  fashion.


                 Table  11.5-2  Dimensionless Groups Used in
                 Nonisothermal  Systems

                 Re  =  ll v /f4 =  ll o v o /p]I  = Reynolds  number
                       o oP
                             ;
                 Pr  = [C>/fc] =  Wai  =  Prandtl  number
                 Gr  = lgP(T ]  - T )l o/A  =  Grashof number
                            '
                               3
                              o
                 o r  —  ^JJLVQ/ K\l  i  -To)}  =  Brinkman  number
                 Pe  = RePr          =  Peclet number
                 Ra  = GrPr          =  Rayleigh  number
                 Ec =  Br/Pr         =  Eckert  number
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