ME 150 – Heat and Mass Transfer
Chap. 16.2: NTU Method for Heat Exchangers
Effectiveness-NTU Method for Heat Exchangers
Maximum possible heat transfer: - Counter flow heat exchanger - Infinitively long - Maximum temperature difference for one fluid: ΔT = Th,i – Tc,i
Prof. Nico Hotz
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ME 150 – Heat and Mass Transfer
Chap. 16.2: NTU Method for Heat Exchangers
Maximum possible heat transfer: Case 1:
m c ⋅ c p ,c < m h ⋅ c p ,h
Counter flow heat exchanger: Infinitively long
qmax = m c ⋅ c p ,c ⋅ ΔTmax = m c ⋅ c p ,c ⋅ (Th ,i − Tc ,i )
Case 2:
m c ⋅ c p ,c > m h ⋅ c p ,h
qmax = m h ⋅ c p ,h ⋅ ΔTmax = m h ⋅ c p ,h ⋅ (Th ,i − Tc ,i )
In other words: qmax = min (m ⋅ c p )⋅ (Th ,i − Tc ,i ) Prof. Nico Hotz
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ME 150 – Heat and Mass Transfer
Chap. 16.2: NTU Method for Heat Exchangers
Effectiveness of heat exchanger: Effectiveness is the actual transferred heat divided by the maximum possible heat transfer. ε=
q qmax
m c ⋅ c p ,c ⋅ (Tc ,o − Tc ,i ) m h ⋅ c p ,h ⋅ (Th ,i − Th ,o ) = = min(m ⋅ c p )⋅ (Th ,i − Tc ,i ) min(m ⋅ c p )⋅ (Th ,i − Tc ,i )
On the other hand, effectiveness is a function of: ⎛ min(m ⋅ c p ) ⎞ ⎜ ⎟ ε = f ⎜ NTU, max(m ⋅ c p ) ⎟⎠ ⎝
with Number of Transfer Units (NTU):
Prof. Nico Hotz
NTU =
U⋅A min(m ⋅ c p ) 3
ME 150 – Heat and Mass Transfer
Chap. 16.2: NTU Method for Heat Exchangers
Effectiveness Relations: Parallel flow HX, Case 2:
m c ⋅ c p ,c > m h ⋅ c p ,h
m h ⋅ c p ,h ⋅ (Th ,i − Th ,o ) m h ⋅ c p ,h ⋅ (Th ,i − Th ,o ) Th ,i − Th ,o ε= = = ( ) ( ) ( ) min m ⋅ c p ⋅ Th ,i − Tc ,i mh ⋅ c p ,h ⋅ Th ,i − Tc ,i Th ,i − Tc ,i
From LMTD method: ⎛ 1 ⎛ Th,o − Tc,o ⎞ 1 ⎞⎟ ⎜ ⎜ ⎟ ln⎜ ⎟ = −U ⋅ A ⋅ ⎜ m ⋅ c + m ⋅ c ⎟ T − T c p ,c ⎠ ⎝ h,i c,i ⎠ ⎝ h p ,h ⎛ min(m ⋅ c p ) ⎞ ⎛ Th,o − Tc,o ⎞ U⋅A ⎟ ⎜ ⎟ ln⎜ =− ⋅ ⎜1 + ⎟ ⎜ min(m ⋅ c p ) ⎝ max(m ⋅ c p ) ⎟⎠ ⎝ Th,i − Tc,i ⎠ Prof. Nico Hotz
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ME 150 – Heat and Mass Transfer
Chap. 16.2: NTU Method for Heat Exchangers
Using the definition of NTU: Th ,o − Tc ,o Th ,i − Tc ,i
⎡ ⎛ min(m ⋅ c p ) ⎞⎤ ⎟⎥ = exp⎢− NTU ⋅ ⎜1 + ⎜ ⎟ ⎢⎣ ⎝ max (m ⋅ c p ) ⎠⎥⎦
With the equation: min(m ⋅ c p ) Tc ,o − Tc ,i = max (m ⋅ c p ) Th ,i − Th ,o
This leads to: Th,o − Tc,o Th,i − Tc,i
⎛ min(m ⋅ c p ) ⎞ ⎟ = 1 − ε ⋅ ⎜1 + ⎜ max(m ⋅ c ) ⎟ p ⎠ ⎝
Prof. Nico Hotz
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ME 150 – Heat and Mass Transfer
Chap. 16.2: NTU Method for Heat Exchangers
Resulting in: ⎡ ⎛ min(m ⋅ c p ) ⎞ ⎛ min(m ⋅ c p ) ⎞⎤ ⎟ = exp⎢− NTU ⋅ ⎜1 + ⎟⎥ 1 − ε ⋅ ⎜1 + ⎜ max (m ⋅ c ) ⎟ ⎜ ⎟ ⎢⎣ p ⎠ ⎝ ⎝ max (m ⋅ c p ) ⎠⎥⎦
And finally, a relation between NTU and effectiveness: ⎡ ⎛ min(m ⋅ c p ) ⎞⎤ ⎟ 1 − exp⎢− NTU ⋅ ⎜1 + ⎜ max (m ⋅ c ) ⎟⎥ p ⎠ ⎥ ⎢⎣ ⎝ ⎦ ε= ⎛ min(m ⋅ c p ) ⎞ ⎜1 + ⎟ ⎜ max (m ⋅ c ) ⎟ p ⎠ ⎝
Parallel flow HX, Case 1:
m c ⋅ c p ,c < m h ⋅ c p ,h
→ Exactly the same result Prof. Nico Hotz
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ME 150 – Heat and Mass Transfer
Chap. 16.2: NTU Method for Heat Exchangers
Effectiveness relations for other configurations: Counter flow HX: ⎡ ⎛ min(m ⋅ c p ) ⎞⎤ ⎟ 1 − exp⎢− NTU ⋅ ⎜1 − ⎜ max (m ⋅ c ) ⎟⎥ p ⎠ ⎥ ⎢⎣ ⎝ ⎦ ε= ⎡ ⎛ min(m ⋅ c p ) ⎞⎤ min(m ⋅ c p ) ⎟⎥ 1− ⋅ exp⎢− NTU ⋅ ⎜1 − ⎜ ⎟ max (m ⋅ c p ) ⎢⎣ ⎝ max (m ⋅ c p ) ⎠⎥⎦
Cross flow HX, unmixed: ⎧⎪ max (m ⋅ c p ) ⎡ ⎛ min(m ⋅ c p ) ⎞ ⎤ ⎫⎪ 0.22 0.78 ⎟ − 1⎥ ⎬ ε = 1 − exp⎨ ⋅ NTU ⋅ ⎢exp⎜ − NTU ⋅ ⎜ max (m ⋅ c p ) ⎟⎠ ⎥⎦ ⎪ ⎢⎣ ⎝ ⎪⎩ min(m ⋅ c p ) ⎭
Prof. Nico Hotz
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ME 150 – Heat and Mass Transfer
Chap. 16.2: NTU Method for Heat Exchangers
NTU Relations: Parallel flow HX: ⎡ ⎛ min(m ⋅ c p ) ⎞⎤ ⎟⎥ ln ⎢1 − ε ⋅ ⎜1 + ⎜ max (m ⋅ c ) ⎟ p ⎠ ⎥ ⎢⎣ ⎝ ⎦ NTU = − ⎛ min(m ⋅ c p ) ⎞ ⎜1 + ⎟ ⎜ max (m ⋅ c ) ⎟ p ⎠ ⎝
Counter flow HX: ⎡ ⎛ min(m ⋅ c p ) ⎞⎤ ln ⎢(ε − 1) ⎜ ε ⋅ − 1⎟ ⎜ max (m ⋅ c ) ⎟⎥ p ⎢ ⎝ ⎠⎥⎦ NTU = ⎣ ⎛ min(m ⋅ c p ) ⎞ ⎜ − 1⎟ ⎜ max (m ⋅ c ) ⎟ p ⎝ ⎠ Prof. Nico Hotz
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ME 150 – Heat and Mass Transfer
Chap. 16.2: NTU Method for Heat Exchangers
Heat transfer by HX:
ε=
q qmax
→ q = ε ⋅ qmax = ε ⋅ min(m ⋅ c p )⋅ (Th,i − Tc ,i )
If the inlet temperatures, both flow rates and specific heat values, the surface area and the overall heat transfer are known, we can calculate the heat transfer between both fluids.
Prof. Nico Hotz
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ME 150 â&#x20AC;&#x201C; Heat and Mass Transfer
Chap. 16.2: NTU Method for Heat Exchangers
NTU Method for Heat Exchangers: The NTU Method is used to evaluate heat exchangers for known inlet temperatures of the fluids and a known geometry of the heat exchanger (outlet temperatures unknown). Possible Procedure to Analyze Heat Exchanger: 1) Calculate overall thermal resistance from known geometry. 2) Calculate NTU from overall thermal resistance. 3) Calculate effectiveness from NTU. 4) Calculate total heat transfer q from effectiveness and inlet temperatures. 5) Calculate outlet temperatures from known inlet temperatures and total heat transfer q.
Prof. Nico Hotz
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ME 150 – Heat and Mass Transfer
Chap. 16.2: NTU Method for Heat Exchangers
Phase Change in Heat Exchanger: Condensing vapor: Phase change leads to constant temperature on hot side, flow direction of hot side is irrelevant. Same result for:
m c ⋅ c p ,c << m h ⋅ c p ,h
Prof. Nico Hotz
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ME 150 – Heat and Mass Transfer
Chap. 16.2: NTU Method for Heat Exchangers
Phase Change in Heat Exchanger (2): Evaporating liquid: Phase change leads to constant temperature on cold side, flow direction of cold side is irrelevant. Same result for:
m c ⋅ c p ,c >> m h ⋅ c p ,h
Prof. Nico Hotz
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ME 150 – Heat and Mass Transfer
Chap. 16.2: NTU Method for Heat Exchangers
Same Fluid on Both Sides of Heat Exchanger: m c ⋅ c p ,c = m h ⋅ c p ,h
→ min (m ⋅ c p ) = max (m ⋅ c p )
Parallel flow HX:
ε = 0.5 ⋅ [1 − exp(− NTU ⋅ 2)]
NTU = −0.5 ⋅ ln(1 − ε ⋅ 2)
Counter flow HX: ε=
NTU 1 + NTU
NTU =
ε 1− ε
Temperature difference ∆T is constant
Prof. Nico Hotz
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ME 150 â&#x20AC;&#x201C; Heat and Mass Transfer
Prof. Nico Hotz
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