This preview shows page 1. Sign up to view the full content.
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
– logmean temperature difference log– EffectivenessNTU 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 multiphase 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 = TH2TC2 ΔTinlet =TH1TC1 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 = TH1TC1 Hot Fluid Cold Fluid ΔToutlet = TH2TC2 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 = TH1TC1 Cold Fluid ΔToutlet = TH2TC2 Area, A Counter flow
Hot Fluid
Fluid Bulk Temperature, TB ΔTinlet = TH1TC2 Cold Fluid Fl ΔToutlet = TH2TC1 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/hft2F 1.1 0.4 2001000 2001500 150300 U W/m2C 6.2 2.3 11005600 11008500 8501700 2555 28280 1040 Plate Glass Window Double Plate Glass Window Steam Condenser Feed water Heater WatertoWater Heat Exchanger Finned Tube Exchanger (WaterAir) 510 Finned Tube Exchanger (SteamAir) 550 GastoGas Exchanger 28 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 Logmean Temperature Difference EffectivenessNTU 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 LogLogmean 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) LogLogmean 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) LogLogmean 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) LogLogmean temperature difference q (ΔT )2 − (Δ T )1 = UA ⎡ Δ T2 ⎤ ln ⎢ ΔT1 ⎥ ⎣ ⎦ (39.12) This term is defined as the logarithmicmean (“logmen”) temperature difference, ΔTlm LogLogmean 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) LogLogmean 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. LongLongmean 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 logmean temperature difference is based on countercounterflow, 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) ...
View
Full
Document
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.
 Spring '09
 DrNesreenGhaddar
 Heat Transfer

Click to edit the document details