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Process  Heat Transfer                                        165


            where the logarithmic-mean diameter, D LM, is defined by



            D LM =  ——————                                              (4.12)
                 In (D 0 /DO

                 Each term in the  denominator  of  Equation 4.11  is  the reciprocal  of a heat-
            transfer  coefficient,  and thus represents a resistance to heat transfer.  The first term
            in  the  denominator  represents  the  resistance  to  heat  conduction  across  a  scale
            formed  on the inside surface  of the tube, where the thickness and the thermal con-
            ductivity of the  scale is rarely known. The thermal conductivity and the thickness
            of  scale  are  not  reported  in the  literature, but  its reciprocal  is  designated by  Rf j,
            the resistance to heat transfer  caused by the tube-side scale, where

            R fi = x f i D 0 /k f i Di                                  (4.13)

            Also,  Rf  0,  the  resistance  to  heat  transfer  (fouling  resistance  or  fouling  factor)
            caused by the shell-side scale is equal to the last term in the denominator of Equa-
            tion4.11.



            R fo =  x f o /k f o                                        (4.14)


                 The  scale thickness will vary with time.  When a heat  exchanger is  first  in-
            stalled, it is clean.  With use the scale thickness increases.  If a fouling resistance is
            specified,  the time required to form the scale is indirectly specified,  usually  1 to  1
            !/  years  [1]. When  this  period  of  time  is  reached,  the  heat  exchanger  must  be
             2
            taken  out  of  service  and  cleaned.  The  longer  the  time  before  cleaning  (service
            time), the greater the required heat-transfer  area and cost of the heat exchanger and
            hence capital cost,  but the cost of cleaning and operating cost will be less.  On the
            other  hand,  if the  service time  is reduced,  the  heat-exchanger  cost  will  decrease,
            but  the  cleaning  cost  will  increase.  Therefore,  there  is  an  optimum  service  time
            which  minimizes  the  total  cost.  This  optimization problem  has been  studied  by
            Crittenden and Khater [26].
                 The second term in the denominator of Equation 4.11 represents the convec-
            tive resistance to heat transfer  caused by the inside  fluid  film  on the scale surface.
            The  third term is the conductive resistance  caused by the tube  wall,  which is usu-
            ally small, because  the thermal conductivity of many metals is large. We will ne-
            glect the conductive resistance  to heat transfer,  unless the thermal  conductivity  is
            very  small  and tube  wall thickness  large.  The  fourth  term is the convective resis-
            tance to heat transfer  of the  outside fluid  film on the scale  surface.  After  substitut-
            ing Equations 4.13  and 4.14 into Equation 4.11,





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