1. Flow Inside Heat Exchangers
Heat Transfer
Flow Inside Heat Exchangers

2. Flow inside Heat Exchangers
In Heat Exchangers, one fluid is cooled while the other fluid is heated without direct contact between fluids.
There are two types of flow inside Heat Exchangers.

3. 1- Parallel Flow 2- Countercurrent Flow (Co current Flow) T1 T1
W,Cp,T1
W,Cp,T2
W,Cp,T2
W,Cp,T1
w,cp,t1
w,cp,t1
w,cp,t2
w,cp,t2 x
temperature x
temperature T1 T1 T2 T2 t2 t2 t1 t1

4. Q=w cp (t2-t1)=W Cp (T1-T2)=Uo Ao ∆tm
Typically in heat exchangers, the heat lost by the hot fluid is gained by the cold fluid through indirect contact (neglecting any heat losses), an overall Energy Balance Equation can be written as:
Q=w cp (t2-t1)=W Cp (T1-T2)=Uo Ao ∆tm
Uo :Overall heat transfer coefficient
Ao :Heat transfer area
∆tm:Average temperature difference
𝟏 𝐔 𝐨 𝐀 𝐨 = 𝟏 𝐡 𝐢 𝐀 𝐢 + 𝐑 𝐰𝐚𝐥𝐥 + 𝟏 𝐡 𝐨 𝐀 𝐨 hi ho
Rwall can be neglected compared with convection resistances.
𝟏 𝐔 𝐨 𝐀 𝐨 = 𝟏 𝐡 𝐢 𝐀 𝐢 + 𝟏 𝐡 𝐨 𝐀 𝐨
𝟏 𝐔 𝐨 = 𝟏 𝐡 𝐢 𝐨 + 𝟏 𝐡 𝐨
Where 𝐡 𝐢 𝐨 is the heat transfer coefficient of the inner fluid based on the outside area

5. The differential rate of heat transfer dQ is dQ=Uo 𝛑 Do dx ∆t
For a differential element inside the exchanger, with a thickness dx, T & t are the mean values of the hot and cold fluids respectively, ∆t=T-t T t dQ dx
The differential rate of heat transfer dQ is
dQ=Uo 𝛑 Do dx ∆t
dQ=±W C p dT=w c p dt t1 t2 T2 T1
Q= Q Qtotal
temperature
∆ 𝐭 𝐚
∆ 𝐭 𝐛
Parallel Flow t1 t2 T2 T1
temperature
∆ 𝐭 𝐚
∆ 𝐭 𝐛
Q= Q Qtotal
Countercurrent Flow

6. t1 t2 T2 T1
Q= Q Qtotal
temperature
∆ 𝐭 𝐚
∆ 𝐭 𝐛 t1 t2 T2 T1
temperature
∆ 𝐭 𝐚
∆ 𝐭 𝐛
Q= Q Qtotal
Parallel Flow
𝐝𝐐=𝐔 𝐨 𝛑 𝐃 𝐨 𝐝𝐱.∆𝐭
Countercurrent Flow
𝐝(∆𝐭) 𝐝𝐐 = 𝐝 𝐓−𝐭 𝐝𝐐 = 𝐝𝐓 𝐝𝐐 − 𝐝𝐭 𝐝𝐐 =± 𝟏 𝐖. 𝐂 𝐩 − 𝟏 𝐰. 𝐜 𝐩 =Constant
𝐝(∆𝐭) 𝐝𝐐 = ∆ 𝐭 𝐛 −∆ 𝐭 𝐚 𝐐 𝐭𝐨𝐭𝐚𝐥 −𝟎
𝐝(∆𝐭) 𝐔 𝐨 𝛑 𝐃 𝐨 𝐝𝐱.∆𝐭 = ∆ 𝐭 𝐛 −∆ 𝐭 𝐚 𝐐 𝐭𝐨𝐭𝐚𝐥
𝐝(∆𝐭) ∆𝐭 = ∆ 𝐭 𝐛 −∆ 𝐭 𝐚 𝐐 𝐭𝐨𝐭𝐚𝐥 𝐔 𝐨 𝛑 𝐃 𝐨 𝐝𝐱
∆𝐭 𝐚 ∆𝐭 𝐛 𝐝(∆𝐭) ∆𝐭 = ∆ 𝐭 𝐛 −∆ 𝐭 𝐚 𝐐 𝐭𝐨𝐭𝐚𝐥 𝐔 𝐨 𝛑 𝐃 𝐨 𝟎 𝐥 𝐝𝐱

7. 𝐐 𝐭𝐨𝐭𝐚𝐥 = 𝐔 𝐨 𝛑 𝐃 𝐨 𝐥 ( ∆ 𝐭 𝐛 −∆ 𝐭 𝐚 𝐥𝐧 ∆ 𝐭 𝐛 ∆ 𝐭 𝐚 )
∆𝐭 𝐚 ∆𝐭 𝐛 𝐝(∆𝐭) ∆𝐭 = ∆ 𝐭 𝐛 −∆ 𝐭 𝐚 𝐐 𝐭𝐨𝐭𝐚𝐥 𝐔 𝐨 𝛑 𝐃 𝐨 𝟎 𝐥 𝐝𝐱
𝐥𝐧 ∆ 𝐭 𝐛 ∆ 𝐭 𝐚 = ∆ 𝐭 𝐛 −∆ 𝐭 𝐚 𝐐 𝐭𝐨𝐭𝐚𝐥 𝐔 𝐨 𝛑 𝐃 𝐨 𝐥
𝐐 𝐭𝐨𝐭𝐚𝐥 = ∆ 𝐭 𝐛 −∆ 𝐭 𝐚 𝐥𝐧 ∆ 𝐭 𝐛 ∆ 𝐭 𝐚 𝐔 𝐨 𝛑 𝐃 𝐨 𝐥
𝐐 𝐭𝐨𝐭𝐚𝐥 = 𝐔 𝐨 𝛑 𝐃 𝐨 𝐥 ( ∆ 𝐭 𝐛 −∆ 𝐭 𝐚 𝐥𝐧 ∆ 𝐭 𝐛 ∆ 𝐭 𝐚 )
𝐐 𝐭𝐨𝐭𝐚𝐥 = 𝐔 𝐨 𝛑 𝐃 𝐨 𝐥 ∆ 𝐭 𝐦
Where ∆ 𝐭 𝐦 is the LMTD: Logarithmic Mean Temperature Difference

8. Example (1)
A hot fluid enters the exchanger at T1=300oF and leaves at T2oF, A cooler fluid enters the exchanger at t1=100oF and leaves at t2oF
a) If T2=200oF and t2=170oF
b) If T2=t2=182oF
c) If T2=110oF and t2=233oF

9. In case of Isothermal Fluids
If a cold fluid is heated from 100oF to 275oF by condensing steam at 300oF, Calculate LMTD for both cases
100
275
300
temperature
Parallel Flow
100
275
300
temperature
Countercurrent Flow