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Heat-Exchangers_Comp - Heat Heat Exchangers 1 Basic Types of Heat Exchangers Heat Exchanger Performance Overview • General principle of operation

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Unformatted text preview: Heat Heat Exchangers - 1 Basic Types of Heat Exchangers Heat Exchanger Performance Overview • General principle of operation • Classes or types of heat exchangers Classes or types of heat exchangers • Overall conductance – General expression for UA – Fouling factors, Rf • Heat exchanger performance – log-mean temperature difference log– Effectiveness-NTU method Effectiveness- General operating principle... Objective • To exchange heat between two fluids across a thermal fl th resistance. resistance. • A general operating equation is is , & & Q = Q( xi , i = 1,2...n) (38.1) Operational parameters Operational Operational parameters • Average temperature difference between hot and cold fluid. fl • Reynolds number of both fluids. • Mixed or unmixed fluids. • Geometry of the flow passages. • Parallel or counter flow. Operational parameters • Condition of the heat transfer surface. • Fluid properties. • Single or multi-phase flow. multi• Volumetric heat capacity of the two fluids. Types of heat exchangers... exchangers... Shell and tube heat exchangers Tube side fluid TB1,i Shell side fluid TB2,i TB2,o TB1,o The internal structure of the baffles on the “shell side” plays a major role on the pressure drop and total heat transfer rate. Temperature Temperature vs. area plots The bulk temperatures of the hot and cold side fluids are usually plotted as a function of area. TB inlet A outlet Boilers Fluid Bulk Temperature, TB Hot Fluid ΔToutlet = TH2-TC2 ΔTinlet =TH1-TC1 Area, A The temperature diagram for a boiler. The hot fluid, which remains at a constant temperature (above boiling) can flow in either direction (parallel or counter) with respect to the cold fluid. Condensers Condensers Fluid Bulk Temperature, TB ΔTinlet = TH1-TC1 Hot Fluid Cold Fluid ΔToutlet = TH2-TC2 Area, A The temperature diagram for a condenser. The cold fluid, which remains at a constant temperature can flow in either direction (parallel or counter) with respect to the cold fluid. Parallel flow Fluid Bulk Temperature, TB Hot Fluid ΔTinlet = TH1-TC1 Cold Fluid ΔToutlet = TH2-TC2 Area, A Counter flow Hot Fluid Fluid Bulk Temperature, TB ΔTinlet = TH1-TC2 Cold Fluid Fl ΔToutlet = TH2-TC1 Area, A Cross flow Fluid flow in tubes Fluid flow across tubes. Flow across the tubes is termed “mixed” flow. Forced Forced convection from a singled heated cylinder in cross flow. Note the extent of the wake when the cylinder in free of effects of nearby cylinders and surfaces. Direction of Direction of cross flow Flow through a tube bank showing the effect of tube placement and the effect of upstream tube on boundary layers on tubes in downstream flow. Cross Cross flow Flow in tubes Cross flow fl between plates is “unmixed.” Overall conductance... The general conductance model Overall conductance TB1 TB2 Two fluids exchange heat through a wall of th h finite finite resistance. Hot fluid TB1 TB2 A1, h1, Re1 A2, h2, Re2 Cold fluid Overall conductance TB1 TB2 x TB1 RC1 RW RC1 TB2 Equivalent thermal circuit element with convective resistances and wall resistance components. & Q1− 2 TB1 RC1 RW RC1 TB2 (38.2) ΔT & Q= = (UA ) total Δ T R & Q = (UA ) total (T B 1 − T B 2 (T B 1 − T B 2 ) & Q= ∑ Ri i ) (38.3) (38.4) Overall conductance - Area for Q Area of fluid “1” TB1 TB2 Area of fluid “2” The area to use in the overall conductance equation, Eq. (38.2) can be chosen as either that for fluid “1” or “2”. Thus, respect to fluid “1”. & = UA1 ΔTavg = ΔTavg Q Rtotal (38.5) Overall Overall Conductance - Typical values Type of Equipment or Device h BTU/h-ft2-F 1.1 0.4 200-1000 200-1500 150-300 U W/m2-C 6.2 2.3 1100-5600 1100-8500 850-1700 25-55 28-280 10-40 Plate Glass Window Double Plate Glass Window Steam Condenser Feed water Heater Water-to-Water Heat Exchanger Finned Tube Exchanger (Water-Air) 5-10 Finned Tube Exchanger (Steam-Air) 5-50 Gas-to-Gas Exchanger 2-8 Overall conductance - U & Δ T avg = UA j Δ T avg Q= R total (38.6) UA j = 1 ∑ Ri j = 1, 2 (38.7) i The largest thermal resistances dominate the overall U and hence the heat transfer rate. Fouling of the surface can introduce large errors into the convective heat transfer coefficients for either hot or cold fluid. Heat Exchanger Performance The Log-mean Temperature Difference Effectiveness-NTU Method Heat exchanger performance factors “Duty”, or Q Area, A, or volume U or ΔP Th The logarithmic mean temperature difference… Operating model Two fluids exchange heat through heat through a wall wall of finite resistance. Hot fluid TB1 TB2 Assumptions: (1) Steady state (2) Constant properties (3) No work. (4) No heat transfer except between fluids A1, h1, Re1 Re A2, h2, Re2 Cold fluid An energy balance (First Law) for an open system comprising the hot and cold fluids for an assumed counter flow situation: TB1,in TB2,out Hot fluid (1) fluid (1) Cold fluid (2) TB1,out TB2,in The control volumes for the hot and cold fluids are contact via a diathermal wall. Thus, contact diathermal Th & & Q1− 2 = −Q2−1 (39.1) TB1,in TB2,out Hot fluid (1) Cold fluid (2) TB1,out TB2,in & & & 0 = Q1 + (m1C p ,1TB1 )in − (m1C p ,1TB1 )out & & Q1 = ( mC )1 ΔTB1 For fluid 2: (39.2) For fluid 1: fluid & & Q 2 = ( m 2 C p , 2 ) ΔT B 2 (39.3) Regardless of the type of heat exchanger, e.g., parallel or cross flow, the following energy balances hold. & & & Q H = (mC )H ΔTB , H = (mC )H (TH 1 − TH 2 ) & & & QC = ( mC ) C ΔTB ,c = (mC )C (TC1 − TC 2 ) & & Q H = − QC (39.4) Note that the sign convention for the First Law analysis of open systems has been maintained with respect to enthalpy flows. Radially lumped energy balances over a differential area element Consider a differential area element, dA, in the exchanger. The heat transfer rate for this area element is expressed terms of the overall conductance for the area element overall conductance for the area element. TB & dQH & dQC dA A TB & dQH For either the hot or the cold fluid, & dQC dA A & dQ = UdA ΔT = UdAd (TH − TC ) = UdA( dT H − dTC ) (39.5) Increment of heat transfer to each of the fluids & ⎧dQ H = ( mC ) H dT H & ⎨& & ⎩ dQC = (mC )C dTC (39.6) From these equations, we can write: & & − dQ dQ H = dTH = & & (mC P )H (mC P )H & & dQC dQ = dTC = & C P )C (mC P )C & (m (39.7) 1⎤ &⎡ 1 + d (ΔT ) = − dQ ⎢ & (m C )H (m C )c ⎥ ⎣& ⎦ ⎡1 1 + d ( Δ T ) = −U Δ T ⎢ & & (m C )c ⎣ (m C )H ⎤ ⎥ dA ⎦ (39.8) This equation applies to any of the generic categories of heat exchangers as the analysis is not dependent on either the relative directions of flow or the internal flow paths. paths. ⎡1 1 + d ( Δ T ) = −U Δ T ⎢ & & (m C )c ⎣ (m C )H d (Δ T ΔT ⎤ ⎥ dA ⎦ (39,8) This equation is easily integrated from the inlet to the outlet. ∫ 2 )=U 1 ⎡ 1 ⎢& ⎣ (m C )H ⎤ 1 − & (m C )C ⎥ ⎦ ∫ dA 2 1 (39.9) where “1” denotes the inlet and “2” denotes the outlet LogLog-mean temperature difference ∫ 2 1 d (Δ T ΔT )= ⎡ 1 −U ⎢ & ⎣ (m C )H 1 + & (m C )C ⎤ ⎥ ⎦ ∫ dA 2 1 (39.9) ⎡1 ⎡ ΔT2 ⎤ 1⎤ ln ⎢ ⎥ = −UA ⎢ (mC ) + (mC ) ⎥ & C⎦ ⎣ ΔT1 ⎦ ⎣& H (39.10) LogLog-mean temperature difference For either the hot or cold fluid, we have from Eqs. (39.1) or (39.2) 1 = ± q (ΔTB ) & (mC ) Thus, Eq. (49.10) can be expressed as ⎡ ΔT ⎤ UA UA ln ⎢ 2 ⎥ = − (TH ,i − TH ,o ) + (TC ,o − TC ,i ) q ⎣ ΔT1 ⎦ [ (39.11) LogLog-mean temperture difference ⎡ ΔT ⎤ UA ln ⎢ 2 ⎥ = − (TH ,i − TH ,o ) + (TC ,o − TC ,i ) q ⎣ ΔT1 ⎦ [ (39.11a) This equation can be rearranged to give ⎡ Δ T2 ⎤ UA ln ⎢ ⎥ = q [(ΔT ) 2 − (Δ T )1 ] ⎣ ΔT1 ⎦ (39.11b) LogLog-mean temperature difference q (ΔT )2 − (Δ T )1 = UA ⎡ Δ T2 ⎤ ln ⎢ ΔT1 ⎥ ⎣ ⎦ (39.12) This term is defined as the logarithmic-mean (“log-men”) temperature difference, ΔTlm LogLog-mean temperature difference q = ΔTlm UA (39.13) q = UA Δ Tlm (Δ T ) 2 − ( ΔT )1 Δ Tlm = ⎡ ΔT2 ⎤ ln ⎢ ΔT1 ⎥ ⎣ ⎦ (39.14) (39 (39.15) LogLog-mean temperature difference Δ Tlm = (Δ T ) 2 − ( ΔT )1 ⎡ ΔT2 ⎤ ln ⎢ ΔT1 ⎥ ⎣ ⎦ (39.15) Note that inlet (“1”) and outlet (“2”) temperature that inlet and outlet temperature differences apply to the heat exchanger regardless of the directions of fluid flow. LongLong-mean temperature difference TB ΔT1 ΔT2 TB ΔT1 ΔT2 Parallel Flow A ΔTlm = (ΔT) 2 − (ΔT )1 ⎡ ΔT2 ⎤ ln ⎢ ΔT1 ⎥ ⎣ ⎦ Counter Flow A (39.15) Correction factors To To correct for flow patterns, e.g., in cross flow and shell and tube heat exchangers, correction factors have been developed developed. q = UAFΔTlm,CF (39.16) where the log-mean temperature difference is based on countercounter-flow, and F takes on values specific to the type of heat exchanger. Correction Correction factors 1 F R= T1, i − T1, 0 T2 , 0 − T2 , i 0.5 P= ΔT1 ΔTi Typical chart for correction factor, F. Numerical values depend on flow pattern and type of heat exchanger. Heat exchanger effectiveness Recall the basic energy balance for a generic heat exchanger. TB1,i TB1,o First Law for the not and & q = −(mC )H ΔTB , H & q = ( m C ) C ΔT B , c TB2,i TB2,o cold fluids: Heat Heat exchanger effectiveness Define the effectiveness of the heat exchanger as the ratio of the actual heat transfer, Q, to the maximum possible possible heat transfer, Qmax. where & Q ε= & Qmax (39.17) && & Q = (mC)H (TH,i −TH,o ) = (mC)C (TC,o −TC,i ) (39.18) Heat exchanger effectiveness The maximum possible heat transfer is defined as that for the fluid with the minimum volumetric heat capacity underundergoing the maximum possible temperature difference that going the maximum possible temperature difference that exists exists in the exchanger. & & Q max = (m C )min Δ T max & = (m C )min (T H , i − T C , i ) (39.19) Thus, effectiveness can be expressed in terms of either the hot or cold fluid, whichever has the minimum volumetric heat capacity. Heat Heat exchanger effectiveness & & Q max = (m C )min Δ T max & = (m C )min (T H , i − T C , i ) (39.19) If the hot fluid has the minimum volumetric heat capacity, & (mC)H (TH,i −TH,o ) εH = & (mC)H (TH,i −TC,i ) = (TH,i −TH,o ) (TH,i −T,Ci ) (39.20) Heat exchanger effectiveness If the cold fluid has the minimum volumetric heat capacity, εC = = & (mC ) C (TC ,o − TC ,i ) & (mC )C (TH ,i − TC ,i ) (TC ,o − TC ,i ) (TH ,i − TC ,i ) (39.21) Analytical expressions are possible for ε based on the fundamental energy balances. Heat Heat exchanger effectiveness Define the number of transfer units, NTU, by the ratio of the overall conductance to the minimum volumetric heat capacit . ity UA NTU = & (mC )min The effectiveness is thus given by, effectiveness is thus given by, (39.22) ε = ε ⎛ NTU , (mC )min (mC ) ⎜ & ⎝ & ⎞ ⎟ max ⎠ (39.23) Effectiveness - NTU From the integral energy balance over the entire heat exchange area, we have ⎡1 ⎡ ΔT ⎤ 1⎤ ln ⎢ 2 ⎥ = −UA⎢ + ⎥ & & ⎣ ΔT1 ⎦ ⎣ (mC )H (mC )C ⎦ which can be written (39.10) ⎡ ΔT ⎤ UA ln ⎢ 2 ⎥ = − & (mC )min ⎣ ΔT1 ⎦ & ⎡ (mC )min ⎤ ⎢1 + & (mC )max ⎥ ⎣ ⎦ (39.24) Effectiveness Effectiveness - NTU ⎡ΔT ⎤ UA ln ⎢ 2 ⎥ = − & (mC )min ⎣ ΔT1 ⎦ & ⎡ (mC )min ⎤ 1+ ⎢ ⎥ & ⎣ (mC )max ⎦ (39.24) ⎡ & ⎛ (mC )min ΔT2 = exp⎢− NTU ⎜1 + ⎜ (mC ) & max ΔT1 ⎢ ⎝ ⎣ ⎞⎤ ⎟⎥ ⎟ ⎠⎥ ⎦ (39.25) The left side can be rearranged in terms of effectiveness and the ratio of the volumetric heat capacities. Effectiveness - NTU & ⎛ Δ T2 ( m C ) min = 1 − ε ⎜1 + ⎜ & Δ T1 ( m C ) max ⎝ ⎞ ⎟ ⎟ ⎠ (39.26) For a parallel flow heat exchanger comprising concentric concentric tubes, we combine Eqs. (39.25) and (39.26) to get: & ⎧ (mC )miin ⎞⎫ 1 − exp ⎨− NTU ⎛1 + ⎜ & (mC )max ⎟ ⎬ ⎝ ⎠⎭ ⎩ ε= & (mC )min 1+ & (mC )max (39.27) Effectiveness Effectiveness - NTU For counter flow (concentric tubes), we have ε= , 1 − Cr exp{ NTU (1 − Cr )} − 1 − exp{ NTU (1 − Cr )} − Cr < 1 (39.28) ε= NTU , NTU Cr = 1 (39.29) Effectiveness - NTU When one of the fluids has an infinite heat capacity, Cr = 0. In this case (e.g., boilers and condensers) the ε - NTU relation for all heat exchangers reduces to the following: ε = 1 − exp( − NTU ), Cr = 0 (39.30) ...
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This note was uploaded on 12/23/2010 for the course MECH 412 taught by Professor Drnesreenghaddar during the Spring '09 term at American University of Beirut.

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