ME 150 – Heat and Mass Transfer
Chap. 17.1: LMTD Method for Heat Exchangers
Log Mean Temperature Difference for Heat Exchangers
m h = m h ,i = m h ,o
q = m h ⋅ c p ,h ⋅ (Th ,i − Th ,o )
m c = m c ,i = m c ,o
q = m c ⋅ c p ,c ⋅ (Tc ,o − Tc ,i )
Prof. Nico Hotz
1
ME 150 – Heat and Mass Transfer
dq = −m h ⋅ c p ,h ⋅ dTh dq = m c ⋅ c p ,c ⋅ dTc
Chap. 17.1: LMTD Method for Heat Exchangers
⎛ 1 1 ⎞⎟ ⎜ dTh − dTc = d (Th − Tc ) = −dq ⋅ + ⎜ m ⋅ c ⎟ ⎝ h p ,h m c ⋅ c p ,c ⎠
On the other hand, the heat transfer between both sides can be calculated as: dq = U ⋅ dA ⋅ (Th − Tc )
Prof. Nico Hotz
2
ME 150 – Heat and Mass Transfer
Chap. 17.1: LMTD Method for Heat Exchangers
Combining the heat transfer equation and the energy balance: ⎛ 1 1 ⎞⎟ ⎜ d (Th − Tc ) = −U ⋅ dA ⋅ (Th − Tc )⋅ + ⎜ m ⋅ c ⎟ ⎝ h p ,h m c ⋅ c p ,c ⎠ ⎛ 1 d (ΔT ) d (Th − Tc ) 1 ⎞⎟ ⎜ = = −U ⋅ + ⋅ dA ⎜ m ⋅ c ⎟ (Th − Tc ) ΔT m ⋅ c c p ,c ⎠ ⎝ h p ,h
After integration: ⎛ 1 ⎛ ΔT2 ⎞ 1 ⎞⎟ U⋅A ⎜ ⎜ ⎟ ln⎜ = − U ⋅ A ⋅ + = − ⋅ [(Th,i − Tc,i ) − (Th,o − Tc,o )] ⎟ ⎜ m ⋅ c ⎟ Δ T m ⋅ c q c p ,c ⎠ ⎝ 1 ⎠ ⎝ h p ,h q = U ⋅ A⋅
ΔTo − ΔTi ln(ΔT2 ΔT1 )
Prof. Nico Hotz
1: x = 0 2: x = L 3
ME 150 – Heat and Mass Transfer
Chap. 17.1: LMTD Method for Heat Exchangers
LMTD Method for Parallel Flow Heat Exchangers
1: x = 0, inlet for cold and hot 2: x = L, outlet for cold and hot
ΔTo − ΔTi ΔT2 − ΔT1 q = U ⋅ A⋅ = U ⋅ A⋅ ln(ΔT2 ΔT1 ) ln(ΔT2 ΔT1 )
Prof. Nico Hotz
ΔT1 = ΔTi = Th ,i − Tc ,i
ΔT2 = ΔTo = Th ,o − Tc ,o
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ME 150 – Heat and Mass Transfer
Chap. 17.1: LMTD Method for Heat Exchangers
LMTD Method for Counter Flow Heat Exchangers
1: x = 0, hot inlet and cold outlet 2: x = L, cold inlet and hot outlet
ΔT2 − ΔT1 q = U ⋅ A⋅ ln(ΔT2 ΔT1 )
ΔT1 = Th ,1 − Tc ,1 = Th ,i − Tc ,o
ΔT2 = Th , 2 − Tc , 2 = Th ,o − Tc ,i
Prof. Nico Hotz
5
ME 150 – Heat and Mass Transfer
Chap. 17.1: LMTD Method for Heat Exchangers
LMTD Method for Cross Flow Heat Exchangers
P= R=
q = U ⋅ A ⋅ F ⋅ ΔTlm, cross
Tc ,o − Tc ,i Th ,i − Tc ,i Th ,i − Th ,o Tc ,o − Tc ,i
ΔTlm, cross =
(T − T )− (T ln[(T − T ) (T h ,i
h ,i
Prof. Nico Hotz
c ,o
c ,o
h ,o h ,o
− Tc ,i )
− Tc ,i )]
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ME 150 – Heat and Mass Transfer
Chap. 17.1: LMTD Method for Heat Exchangers
Overall Heat Transfer: Tcold flow Cold fluid:
Rtot = Rcold , conv + Rwall , cond + Rhot , conv
Rcold,conv Tcold wall
Wall:
Rwall,cond Thot wall
Hot fluid:
Rtot =
Rhot,conv
Thot flow
1 t 1 + + hcold ⋅ Acold k wall ⋅ Ac hhot ⋅ Ahot
U ⋅ Awet =
1 Rtot
→ qtot = U ⋅ Awet ⋅ (Thot , flow − Tcold , flow )
Prof. Nico Hotz
7
ME 150 – Heat and Mass Transfer
Chap. 17.1: LMTD Method for Heat Exchangers
Log Mean Temperature Difference Method: The LMTD Method is used to design heat exchangers for known inlet and outlet temperatures of the fluids and a known geometry of the heat exchanger. Possible Procedure to Design Heat Exchanger: 1) Determine known or specified inlet and outlet temperatures. 2) Calculate total heat transfer from inlet and outlet temperatures and fluid properties. 3) Calculate LMTD using formula for the given heat exchanger configuration (parallel, counter, cross flow). 4) Calculate overall thermal resistance using q and LMTD. 5) Calculate geometry from overall thermal resistance and heat transfer coefficients.
Prof. Nico Hotz
8
ME 150 – Heat and Mass Transfer
Prof. Nico Hotz
9