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  Optimum design of double pipe heat exchanger Prabhata K. Swamee a,* , Nitin Aggarwal b , Vijay Aggarwal c a Department of Civil Engineering, Indian Institute of Technology, Roorkee, Roorkee 247 667, India b Department of Chemical Engineering, New Jersey Institute of Technology, NJ 07032, USA c UOP, Unitech Trade Center, Sector 43, Block C, Sushant Lok Phase-1, Gurgaon, 122 001 Haryana, India Received 31 March 2006; received in revised form 10 May 2007Available online 31 December 2007 Abstract Heat exchangers are used in industrial processes to recover heat between two process fluids. Although the necessary equations for heattransfer and the pressure drop in a double pipe heat exchanger are available, using these equations the optimization of the system cost islaborious. In this paper the optimal design of the exchanger has been formulated as a geometric programming with a single degree of difficulty. The solution of the problem yields the optimum values of inner pipe diameter, outer pipe diameter and utility flow rate to beused for a double pipe heat exchanger of a given length, when a specified flow rate of process stream is to be treated for a given inlet tooutlet temperature.   2007 Elsevier Ltd. All rights reserved. Keywords:  Design; Economy; Geometric programming; Heat exchanger; Optimization 1. Introduction Heat transfer equipment is defined by the function itfulfills in a process. The objective of any such equipmentis to maximize the heat transferred between the two flu-ids. However, the problem that occurs is that the param-eters which increase the heat transfer also increase thepressure drop of the fluid flowing in a pipe whichincreases the cost of pumping the fluid. Therefore, adesign which increases the heat transferred, but simulta-neously could keep the pressure drop of the fluid flowingin the pipes to permissible limits, is very necessary. Acommon problem in industries is to extract maximumheat from a utility stream coming out of a particularprocess, and to heat a process stream. A solution toextract the maximum heat could have been to increasethe heat transfer area or to increase the coolant flow ratebut both the solutions increase the cost of pumping soincreasing these parameters without pressure drop con-siderations is not advisable. Traditional design methodof heat exchangers involves the consideration of all thedesign variables with a laborious procedure of trial anderror, taking all possible variations into consideration.Though this time consuming procedure can be reducedsomewhat by making some reasonable assumptions asdescribed by [6], but still no convenient method has beendeveloped for optimal design of double pipe heatexchangers. In other optimum design methods, such asLagrange multiplier method, the optimum results areagain obtained in a long time by changing one variableat a time and using a trial-error or a graphical method.In the current literature (for example, [8]), focus is onoptimizing the area of the heat exchanger irrespective of the different flow rates of the utility that can be used.Using this pressure drop is not minimized to the fullestextent. This fact can be avoided through the designmethod discussed in the paper. We have considered thedesign of a double pipe heat exchanger in which its costis optimized by considering three main parameters – the 0017-9310/$ - see front matter    2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijheatmasstransfer.2007.10.028 * Corresponding author. Tel.: +91 1332 285420; fax: +91 1332 273560. E-mail addresses:  swamifce@iitr.ernet.in (P.K. Swamee), nitinagg.iitr@gmail.com (N. Aggarwal), vijayaggarwal2@gmail.com (V. Aggar- wal).www.elsevier.com/locate/ijhmt  Available online at www.sciencedirect.com International Journal of Heat and Mass Transfer 51 (2008) 2260–2266  inner and outer diameter of the heat exchanger and theflow rate of the utility. The design of the exchangerhas been formulated as a geometric programming witha single degree of difficulty. It is assumed that the flowrate, the inlet and the required outlet temperature of the process fluid and the inlet temperature of the utilityare known for the specific design of the exchanger. 2. Analytical considerations  2.1. Equations for heat transfer coefficients for fluids in pipes Sieder and Tate [3] gave the following equation for bothheating and cooling of a number of fluids in pipes: h i  ¼  0 : 027 d  k  p  Re 0 : 8p C  p l p k  p  13 l p l wp ! 0 : 14 ð 1 Þ where  h i  is the heat transfer coefficient at the inner surfaceof inner pipe;  d   the inner pipe diameter (see Fig. 1);  k  p  thethermal conductivity of the fluid flowing in the inner pipe; C  p  the specific heat of fluid flowing in inner pipe;  l p  the vis-cosity of fluid flowing in the inner pipe;  l wp  the viscosity of fluid in the inner pipe at wall temperature; and  Re p  theReynolds number for inner pipe given by  h i  d  D Inner pipeOuter pipe  h io Fig. 1. Cross sectional view of the double pipe heat exchanger. Nomenclature NotationA  constant (J/m 0.2 s K) a p  flow area of inner pipe (m 2 )  Re 0 a  Reynolds number for pressure drop (Nondimen-sional) A s  surface area for the heat transfer (m 2 ) a a  flow area of annulus (m 2 ) B   constant (J/s 0.2 m 0.2 kg 0.8 K) C   constant (kg m 2.75 /s 2 ) C  p  specific heat of fluid flowing in pipe (J/kg K) d   diameter of inner pipe (m) D  diameter of outer pipe (m) D e  equivalent diameter (m) E   constant (m 2.75 /s 0.25 kg 0.75 ) F   constant (m 2 /kg)  f   friction factor (Nondimensional)  g   gravitational acceleration (m/s 2 ) G  a  mass velocity of fluid in the annulus (kg/m 2 s) G  p  mass velocity of fluid in the inner pipe (kg/m 2 s) h i  heat transfer coefficient of inner pipe (J/m 2 s K) h o  heat transfer coefficient of annulus (J/m 2 s K) k  a  thermal conductivity of the fluid flowing in theannulus (J/s mK) k  p  thermal conductivity of the fluid flowing in theinner pipe (J/s mK) L  total pipe length (m) l  hp  length of one hairpin (m) m  mass flow rate of fluid flowing in the inner pipe(kg/s) Q  heat transfer rate (J/s) Q * required heat transfer rate (J/s) Re a  Reynolds number for annulus (–) Re p  Reynolds number for inner pipe (–) T   temperature of the process stream (K) t  temperature of the utility (K) U  c  overall clean coefficient (J/m 2 s K) w  mass flow rate of fluid flowing in the annulus(kg/s) x D 2   d  2 (m 2 )  y D  +  d   (m) D T   temperature difference (K) l p  viscosity of fluid flowing in the inner pipe (kg/ms) l a  viscosity of fluid flowing in the annulus(kg/ms) l wp  viscosity of fluid in the inner pipe at wall tem-perature (kg/ms) l wa  viscosity of fluid in the annulus at wall temper-ature (kg/ms) q  fluid mass density (kg/m 3 ) Suffixes 1 inlet2 outleta annulusio outer surface of inner pipeLM log meanp inner pipe P.K. Swamee et al./International Journal of Heat and Mass Transfer 51 (2008) 2260–2266   2261   Re p  ¼  d G  p l p ð 2 Þ wherein  G  p  is the mass velocity of fluid in the inner pipegiven by G  p  ¼  ma p ð 3 Þ wherein  m  is the mass flow rate of fluid flowing in the innerpipe; and  a p  the flow area of inner pipe given by a p  ¼  p 4 d  2 ð 4 Þ Combining Eqs. (1)–(4), the following equation isobtained: h i  ¼  Am 0 : 8 d  1 : 8  ð 5 Þ where  A  ¼  0 : 032756 k  p 1 l p ! 0 : 8 C  p l p k  p  13 l p l wp ! 0 : 14 ð 6 Þ It is assumed that the process stream is pumped in the outerpipe and its flow rate is a known quantity, thus constant fora given problem. However the utility is taken in the innerpipe and its flow rate can be varied so that the requiredheat transfer is achieved. The thickness of the inner andthe outer pipes is also considered negligible with respectto their diameters. Further, as all the other variables arethe fluid properties, they are constant for a given fluid.Thus, the heat transfer coefficient of the inner pipe isdependent on its diameter and the flow rate of the utilityonly for a heat exchanger of given length.The equivalent diameter  D e  for heat transfer in the outerpipe is given by  D e  ¼  D 2   d  2 d   ð 7 Þ where  D  is the diameter of outer pipe (See Fig. 1). Using [3] equation for calculating the heat transfer coefficient of theouter pipe one gets h io  ¼  0 : 027  D e k  a  Re 0 : 8a C  pa l a k  a  13 l a l wa  0 : 14 ð 8 Þ where  h io  is the heat transfer coefficient at the outer surfaceof the inner pipe;  k  a  the thermal conductivity of the fluidflowing in the annulus;  C  pa  the specific heat of fluid flowingin annulus;  l a  the viscosity of fluid flowing in the annulus; l wa  the viscosity of fluid in the annulus at outer pipe walltemperature; and  Re a  the Reynolds number for the annu-lus, given by  Re a  ¼  D e G  a l a ð 9 Þ where  G  a  is the mass velocity of fluid in the outer pipe,given by G  a  ¼  wa a ð 10 Þ wherein  w  is the mass flow rate of fluid flowing in the annu-lus; and  a a  the flow area of annulus, given by a a  ¼  p 4 ð  D 2   d  2 Þ ð 11 Þ Further using Eqs. (7)–(11) h io  ¼  Bd  0 : 2  x  ð 12 Þ where  x  ¼  D 2   d  2 ð 13 Þ  B  ¼  0 : 032756 k  a w l a  0 : 8 C  pa l a k  a  13 l a l wa  0 : 14 ð 14 Þ Therefore, the heat transfer coefficient of the outer pipe, fora given length of heat exchanger, depends only upon thediameters of both the pipes, other variables being constantfor the given process stream.  2.2. Equation for pressure drop in the pipes The pressure drop allowance in an exchanger is the sta-tic fluid pressure which may be expended to drive the fluidthrough the exchanger. The pressure that is supplied for thecirculation of a fluid should overcome frictional lossescaused by connecting exchangers in series and the pressuredrop in the exchangers itself. The pressure drop in pipescan be computed from the Darcy-Weisbach equation.Therefore, pressure drop  D  p p  in the length  L  of the innerpipe is given by D  p  p  ¼  f  p  LG  2p 2 q p d   ð 15 Þ where  f  p  is the friction factor for the inner pipe; and  q p  themass density of the inner fluid. Assuming inner pipe to besmooth,  f  p is given by Blasius equation  f  p  ¼  0 : 316  Re 0 : 25p 2 : 1    10 3 <  Re p  <  10 5 ð 16 Þ Using (2)–(4), (15) and (16), the pressure drop is D  p  p  ¼ 0 : 24113 l 0 : 25p  m 1 : 75  L q p d  4 : 75  ð 17 Þ The power required by the pump to overcome the pressuredrop is  P  p  ¼  m q p D  p  p  ð 18 Þ Rewriting Eq. (18) using Eq. (17)  P  p  ¼ 0 : 24113 l 0 : 25p  m 2 : 75  L q 2p d  4 : 75  ð 19 Þ Similarly, the pressure drop in the annulus,  D  p a  is 2262  P.K. Swamee et al./International Journal of Heat and Mass Transfer 51 (2008) 2260–2266   D  p  a  ¼  f  a G  2a  L 2 q a  D 0 e ð 20 Þ where  q a  is the mass density of the outer fluid;  D 0 e  ¼  theequivalent diameter for flow resistance, given by  D 0 e  ¼  D    d   ð 21 Þ  f  a  is the friction factor for the outer pipe; given by Blasiusequation  f  a  ¼  0 : 316 ð  Re 0 a Þ 0 : 25  2 : 1   10 3 6  Re 0 a  6 10 5 ð 22 Þ wherein  Re 0 a  ¼  Reynolds number, given by  Re 0 a  ¼  D 0 e G  a l a ð 23 Þ Combining Eqs. (10) and (11) the mass velocity of fluid inthe outer pipe is G  a  ¼  4 w p ð  D 2   d  2 Þð 24 Þ Putting Eqs. (21)–(24) in Eq. (20) D  p  a  ¼  0 : 24113 l 0 : 25a  y  1 : 25 w 1 : 75  L q a  x 3  ð 25 Þ where  y   ¼  D  þ  d   ð 26 Þ The power required to pump the fluid against this pressuredrop is  P  a  ¼  w q a D  p  a  ð 27 Þ Rewriting Eq. (27) using Eq. (25)  P  a  ¼  0 : 24113 l 0 : 25a  y  1 : 25 w 2 : 75  L q 2a  x 3  ð 28 Þ Adding Eqs. (17) and (25), the overall pressure drop in theheat exchanger is given by D  p   ¼ 0 : 24113 l 0 : 25p  m 1 : 75  L q p d  4 : 75  þ  0 : 24113 l 0 : 25a  y  1 : 25 w 1 : 75  L q a  x 3  ð 29 Þ where entrance and exit losses have been neglected.Adding Eqs. (19) and (28) the total power  P  , which hasto be expended to drive the fluid through the exchanger, is  P   ¼  Cm 2 : 75 d  4 : 75  þ  E  x 3  ð 30 Þ where C   ¼ 0 : 24113 l 0 : 25p  L q 2p ð 31 Þ  E   ¼  0 : 24113 l 0 : 25a  w 2 : 75  y  1 : 25  L q 2a ð 32 Þ  2.3. Equation for the heat transferred between the two pipes Let  T  1  and  T  2  be the inlet and the required outlet tem-perature of the process stream and  t 1  and  t 2  be the inletand the assumed outlet temperature of the utility. SeeFig. 2. The integrated steady state modification of Fou-rier’s general equation is Q  ¼  U  c  A s D T  LM  ð 33 Þ where  Q  is the heat transferred between the fluids per unittime;  U  c  the overall clean coefficient;  A s  the surface area forthe heat transfer, given by  A s  ¼  p d  L  ð 34 Þ and  D T  LM  = log mean temperature difference assumingcounterflow D T  LM  ¼ ð T  1    t  2 Þ  ð T  2    t  1 Þ ln ½ð T  1    t  2 Þ = ð T  2    t  1 Þ ð 35 Þ It is assumed that the thickness of the pipes is negligiblysmall as compared to the inner diameter of the inner pipe.Overall clean coefficient  U  c  can be obtained independentlyof the Fourier equation from the two heat transfer coeffi-cients. Neglecting pipe wall resistance the following equa-tion is obtained:1 U  c ¼  1 h i þ  1 h 0 ð 36 Þ Using Eqs. (34)–(36), (33) changes to 1 Q  ¼  1 h i þ  1 h io   1 p d  L D T  LM ð 37 Þ Using Eqs. (5), (12), (37) changes to1 Q  ¼  d  1 : 8  Am 0 : 8  þ  x Bd  0 : 2   1 p d  L D T  LM ð 38 Þ Also, because heat is conserved, the heat lost by the hotterstream is equal to the heat gained by the coolant. Hence, Q  ¼  mC  p ð t  1    t  2 Þ ð 39 Þ wmt  2 T  1 t 1 mwT  2 Fig. 2. Four double pipe heat exchangers in series. P.K. Swamee et al./International Journal of Heat and Mass Transfer 51 (2008) 2260–2266   2263
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