Surface heat transfer area over volume 𝛼

MEP 460 Heat Exchanger design

King Abdulaziz University

Mechanical Engineering Department

March 2019

Ch. 10 Compact Heat Exchangers

Ch. 10 Compact Heat Exchangers

1-Introduction

2-Tube-fin heat exchangers 3-Plate-fin heat exchangers 4-Examples

1- Introduction

Compact heat exchangers

Surface heat transfer area over volume 𝛼

Tube fin compact heat exchangers

Tube fin compact heat exchangers

Non-circular tubes

2- Tube fin heat exchangers

continuous fins on

flat tubes Continuous fins

on circular tube

Circular fins on circular tubes

Heat transfer and pressure drop for tube fin heat exchangers

1

𝑈_𝑜𝐴_𝑜 = 1

ℎ_𝑖𝜂_𝑖𝐴_𝑖 + 𝑅_𝑓𝑖

𝜂_𝑖𝐴_𝑖 + 𝑅_𝑤 + 𝑅_𝑓𝑜

𝜂_𝑜𝐴_𝑜 + 1 ℎ_𝑜𝜂_𝑜𝐴_𝑜

For h_o outside (gas) heat transfer coefficient use

Kays &

London book

in compact heat exchangers 𝜂_𝑜 = 1 − 𝐴_𝑓

𝐴 (1 − 𝜂_𝑓)

Overall surface efficiency 𝜂_𝑜

𝜂_𝑓 is the fin efficiency

Definitions

Frontal area A

_fr

Free Flow area A

_min

= A

_ff

Fin area/total area=A

_f

/A

_o

𝜎 = 𝐹𝑟𝑒𝑒 𝑓𝑙𝑜𝑤 𝑎𝑟𝑒𝑎

𝐹𝑟𝑜𝑛𝑡𝑎𝑙 𝑎𝑟𝑒𝑎 = 𝐴_𝑓𝑓 𝐴_𝑓𝑟

𝛼 = Surface area/Volume Hydraulic diameter D

_h

Mass velocity G=  u

_max

Definitions

Colburn j

_H

factor 𝑗

_𝐻

= 𝑆𝑡𝑃𝑟

^{2 3Τ}

𝑆𝑡 = 𝑁𝑢

𝑅𝑒 𝑃𝑟 = ℎ

𝜌𝐶_𝑝𝑉_𝑚𝑎𝑥 = ℎ 𝐶_𝑝𝐺

Mass velocity [kg/(m

²

.s)

^{𝐺 = 𝜌𝑉}𝑚𝑎𝑥 = 𝜌𝑉𝐴_𝑓𝑟

𝐴_𝑓𝑓 = 𝑚ሶ

𝐴_𝑓𝑓 = 𝑚ሶ 𝜎𝐴_𝑓𝑟

Stanton Number

𝑅𝑒 = 𝐺𝐷

_ℎ

/𝜇

Δ𝑃 = 𝑓 𝐿

𝐷 𝜌 𝑉²

Friction coefficient f

2

Major Pressure drop

𝑣_𝑚 = 𝑣_𝑖 + 𝑣_𝑜 2 𝐴

𝐴_𝑓𝑓 = 𝛼𝑉 𝜎𝐴_𝑓𝑟 A: heat transfer area

A_fr Frontal area A_ff Free flow area

𝜎 = 𝐹𝑖𝑛 𝑎𝑟𝑒𝑎

𝑇𝑜𝑡𝑎𝑙 𝑎𝑟𝑒𝑎 = 𝐴_𝑓 𝐴

𝛼 = 𝐴 𝑉

Pressure drop gas side

𝑣

_𝑖

specifc volume at inlet 𝑣

_𝑜

specific volume at outlet

𝑣

_𝑚

mean specific volume =

^{𝑣𝑖+𝑣𝑜}

2

The above equation for pressure drop can be also written in terms of densities instead of specific volume

Surface information (CF-7.0-5/8J)

Circular fin on circular tubes

From Incropera 6^th edition

Typical data for tube fin heat exchangers (8.0-3/8T)

Continuous fin on circular tubes

𝐹𝑖𝑛 𝑎𝑟𝑒𝑎

𝑇𝑜𝑡𝑎𝑙 𝑎𝑟𝑒𝑎 = 𝐴

_𝑓

𝐴

𝛼 = 𝛽 = 𝐴 Surface density 𝑉

Surface information

Hydraulic diameter D_h

𝜎 = 𝐹𝑟𝑒𝑒 𝑓𝑙𝑜𝑤 𝑎𝑟𝑒𝑎

𝐹𝑟𝑜𝑛𝑡𝑎𝑙 𝑎𝑟𝑒𝑎 = 𝐴_𝑓𝑓 𝐴_𝑓𝑟

Surface information

1

𝑈_𝑜𝐴_𝑜 = 1

ℎ_𝑖𝜂_𝑖𝐴_𝑖 + 𝑅_𝑓𝑖

𝜂_𝑖𝐴_𝑖 + 𝑅_𝑤 + 𝑅_𝑓𝑜

𝜂_𝑜𝐴_𝑜 + 1 ℎ_𝑜𝜂_𝑜𝐴_𝑜

1

𝑈_𝑜 = 1

ℎ_𝑖( Τ𝐴_𝑖 𝐴_𝑜) + 𝐴_𝑜𝑙𝑛( Τ𝑟_𝑜 𝑟_𝑖)

2𝜋𝑘𝐿 + 1 ℎ_𝑜𝜂_𝑜

Need to know the heat transfer area ratio A_i/A_o

Evaluating overall heat transfer coefficient

Neglecting fouling resistances

𝑅_𝑤 = ln( Τ𝑟_𝑜 𝑟_𝑖) 2𝜋𝑘𝐿

1

𝑈_𝑜 = 1

ℎ_𝑖( Τ𝐴_𝑖 𝐴_𝑜) + 𝐷_𝑖𝑙𝑛( Τ𝑟_𝑜 𝑟_𝑖)

2𝑘 ( Τ𝐴_𝑖 𝐴_𝑜) + 1 ℎ_𝑜𝜂_𝑜

Ratio of inside to outside heat transfer area

𝐴_𝑖 = 𝜋𝐷_𝑖𝐿 𝐴_𝑜,𝑝 = 𝜋𝐷_𝑜𝐿

𝐴_𝑜 = 𝐴_𝑢𝑓 + 𝐴_𝑓 = 𝐴_𝑜,𝑝 + 𝐴_𝑓 𝐴_𝑖

𝐴_𝑜,𝑝 = 𝐷_𝑖 𝐷_𝑜

𝐴_𝑜,𝑝 = 𝐴_𝑜 − 𝐴_𝑓 𝐴_𝑖

𝐴_𝑜 = 𝐷_𝑖

𝐷_𝑜 ∗ 1 − 𝐴_𝑓 𝐴_𝑜 𝐴_𝑖 = 𝐷_𝑖

𝐷_𝑜 𝐴_𝑜,𝑝

D

_o

D

_i

Fins

Inside heat transfer area Outside heat transfer area without fins

Neglecting the area occupied by fins. i.e. A_uf=A_op

 h

_i

is given

 From frontal area A

_fr

,  and mass flow rate get G and Re

_Dh

 Get j from Kays & London graphs and h

_o

 Get fin efficiency for circular fins on circular pipe

 Get U

_o

value

 Knowing q and q

_max

get the effectiveness 

 Knowing C

_r

and the effectiveness  get NTU

 From NTU=UA/C

_min

calculate the heat transfer area A

_o

 Calculate the volume of the heat exchanger using𝛼 = 𝛽 =

^𝐴𝑜

 Get the depth of the heat exchanger L form V=A

𝑉 _fr

L

 Calculate the number of rows of tubes

Example 11.6

Surface: 8.0-3/8 T

Continuous fins on circular tubes

Surface CF-8.72(c) Circular fins on circular tubes

Circular fin on circular tubes

CF-8.7-5/8J

Heat transfer

factor j and friction coefficient f for

some tube-fin and plate- fin surface

Ref.: Kays & London

Continuous fin with flat tube

Some of the data for plate-fin and tube fin compact heat exchangers

Δ𝑝 = 𝐺²

2𝜌_𝑖 4𝑓 𝐿 𝐷_ℎ

𝜌_𝑖

𝜌 + 1 + 𝜎² 𝜌_𝑖

𝜌_𝑜 − 1

Δ𝑝 = 𝐺²

2𝜌_𝑖 𝑓 𝐴 𝐴_𝑚𝑖𝑛

𝜌_𝑖

𝜌 + 1 + 𝜎² 𝜌_𝑖

𝜌_𝑜 − 1 𝐴

𝐴_𝑚 = 4𝐿

𝐷_ℎ = ℎ𝑒𝑎𝑡 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟 𝑎𝑟𝑒𝑎 𝑚𝑖𝑛. 𝑓𝑙𝑜𝑤 𝑎𝑟𝑒𝑎

Pressure drop (Gas side)

A_min min. flow area

A =heat transfer area

P=perimeter A=heat transfer area=P*L

D_h= 4A_min/P

𝐷_ℎ = 4𝐴_𝑚𝑖𝑛 𝑃

𝐿

𝐿 = 4𝐴_𝑚𝑖𝑛𝐿 𝐴

𝐴

𝐴_𝑚𝑖𝑛 = 4𝐿 𝐷_ℎ L

3- Plat fin compact heat exchangers

Kays & London heat transfer and pressure drop data for plate-fin and tube-fin heat exchangers

Ch. 9

Ch. 9 Kays & London

Plain fins

Ch. 9 Kays & London

Strip fins

Ch. 9 Kays & London

Louvered fins

Ch. 9 Kays & London

Ch. 9 Kays & London

Circular tubes, continuous fins Flat tubes continuous fins

Ch. 9 Kays & London

Circular tubes- circular fins

Ch. 9 Kays & London

Flow inside circular and flattened tubes

Δ𝑝 = 𝐺²

2𝜌_𝑖 𝑘_𝑐 + 1 − 𝜎 + 2 𝜌_𝑖

𝜌_𝑜 − 1 + 𝑓 𝐴 𝐴_𝑚𝑖𝑛

𝜌_𝑖

𝜌 − 1 − 𝑘_𝑒 − 𝜎² 𝜌_𝑖 𝜌_𝑜

Entrance Exit

Acceleration

Friction Major loss

Pressure drop for plat-fin heat exchanger

1

𝜌 = 1 2

1

𝜌_𝑖 + 1 𝜌_𝑜 Average density can be found using

Typical data for plate-fin compact heat exchanger

Surface tabulated data for plate-fin compact heat exchangers

Gas

Air

Surface 1 Surface 2

Hydaulic diameter D_h1 Plate spacing b₁

Fin thickness 𝛿_𝑓1

Area/ volume between plate, 𝛽₁ Fin area/heat transfer area, 𝜔₁ Length of the fin, 𝑙_𝑓1

₂ =A₂/V

Hydaulic diameter D_h2 Plate spacing b₂

Fin thickness 𝛿_𝑓2

Area/ volume between plate, 𝛽₂ Fin area/heat transfer area, 𝜔₂ Length of the fin, 𝑙_𝑓2

₁=A₁/V

A_fr1

A_fr2 Plate-fin compact heat exchanger (Gas-to-Gas HX)

Calculate

1-Number of passes N_p

2-Calculate volume between plates for side 1 and side 2

3-Calculate heat transfer area A₁, and A₂ 4-Calculate A_min1, A_min2

5-Calculate G₁, and G₂ 6-Get j₁,j₂, f₁, f₂

7-Calculate h₁ and h₂

8-Calculate η_f1, η_f2, η₀₁,η₀₂ 9-Calculate U value

10-Calculate Cr and NTU, then get 

11-Calculate outlet temperatures 12-Calculate pressure drops

for both sides

L1 L2

L3

𝐿_𝑐 = 𝑁_𝑝𝑏₁ + 𝑁_𝑝 + 1 𝑏₂ + 2 𝑁_𝑝 + 1 𝑎

Assuming the number of passes in one side is N_p and N_p+1 on the other side, then the common edge length can be written in terms the pate spacing's b₁, b₂ and the plate thickness a as follows

Calculating the heat transfer areas for side (1) and side (2)

𝑁_𝑝 = 𝐿_𝑐 − 𝑏₂ − 2𝑎 𝑏₁ + 𝑏₂ + 2𝑎

Volume between the plates for side (1) and side (2)

𝑉_𝑝1 = 𝐿₁ ∗ 𝐿₂ ∗ 𝑁_𝑝 ∗ 𝑏₁ 𝑉_𝑝2 = 𝐿₁ ∗ 𝐿₂ ∗ 𝑁_𝑝 ∗ 𝑏₂

Utilizing the relation between  and the volume between the plates

𝛽₁ = 𝐴₁

𝑉_𝑝1 𝛽₂ = 𝐴₂ 𝑉_𝑝2

If ₁ and ₂ are calculated based on b₁, b₂ a, and  , then one can easily find the heat

transfer areas A₁ and A₂

𝛼₁ = 𝐴₁

𝑉 𝛼₂ = 𝐴₂ 𝑉

Where V is the total volume of the heat exchanger

𝜎₁ = 𝐴_{𝑚𝑖𝑛,1}

𝐴_𝑓𝑟,1 = 𝐴_{𝑚𝑖𝑛,1}𝐿_𝑝1

𝐴_𝑓𝑟,1𝐿_𝑝,1 = 𝐴₁ 𝐷_ℎ,1Τ4

𝑉 = 𝑉₁𝛽₁ 𝐷_ℎ,1Τ4 𝑉

𝑉 = 𝐿₁ ∗ 𝐿₂[𝑏₁𝑁_𝑝 + 𝑏₂(𝑁_𝑝+1) + 2𝑎 ∗ (𝑁_𝑝 + 1)]

𝜎₁ = 𝑚𝑖𝑛. 𝑓𝑙𝑜𝑤 𝑎𝑟𝑒𝑎 𝑓𝑟𝑜𝑛𝑡𝑎𝑙 𝑎𝑟𝑒𝑎

𝛽 = ℎ𝑒𝑎𝑡 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟 𝑎𝑟𝑒𝑎 𝑉𝑜𝑙𝑢𝑚𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡ℎ𝑒 𝑝𝑙𝑎𝑡𝑒𝑠

𝑉 = 𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑡ℎ𝑒 ℎ𝑒𝑎𝑡 𝑒𝑥𝑐ℎ𝑎𝑛𝑔𝑒𝑟

𝜎₁ = 𝑉₁𝛽₁ 𝐷_ℎ,1Τ4

𝐿₁ ∗ 𝐿₂ [𝑏₁𝑁_𝑝 + 𝑏₂(𝑁_𝑝+1) + 2𝑎 ∗ (𝑁_𝑝 + 1)] = 𝐿₁ ∗ 𝐿₂ ∗ 𝑏₁ ∗ 𝑁_𝑝 ∗ 𝛽₁ 𝐷_ℎ,1Τ4

𝐿₁ ∗ 𝐿₂ [𝑏₁𝑁_𝑝 + 𝑏₂(𝑁_𝑝+1) + 2𝑎 ∗ (𝑁_𝑝 + 1)]

𝜎₁ = 𝑏₁𝛽₁ 𝐷_ℎ,1Τ4 𝑏₁ + 𝑏₂ + 2𝑎

𝛼₁ = 𝐴₁

𝑉 = 𝐴₁

𝐴_𝑓𝑟,1𝐿₁ = 𝐴₁Τ𝐿₁

𝐴_𝑓𝑟,1 = 4𝐴_{𝑚𝑖𝑛,1}Τ𝐷_ℎ,1

𝐴_𝑓𝑟,1 = 4𝜎₁

𝐷_ℎ,1 = 𝑏₁𝛽₁ 𝑏₁ + 𝑏₂ + 2𝑎

𝐿_𝑝1 𝑓𝑙𝑜𝑤 𝑝𝑎𝑠𝑠𝑎𝑔𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 𝑓𝑜𝑟 𝑠𝑖𝑑𝑒 (1)

𝑏₁ 𝑏₂ 𝑎𝑟𝑒 𝑡ℎ𝑒 𝑝𝑙𝑎𝑡𝑒 𝑠𝑝𝑎𝑐𝑖𝑛𝑔 𝑓𝑜𝑟 𝑠𝑖𝑑𝑒 1 𝑎𝑛𝑑 𝑠𝑖𝑑𝑒 2

Deducing the relation for ₁ and ₂

𝑁_𝑝 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑎𝑠𝑠𝑒𝑠 𝐷_ℎ = 4𝐴_𝑚𝑖𝑛

𝑃 𝐿

𝐿 = 4𝐴_𝑚𝑖𝑛𝐿 𝐴

𝜎₂ = 𝑏₂𝛽₂ 𝐷_ℎ,2Τ4 𝑏₁ + 𝑏₂ + 2𝑎

𝛼₂ = 𝐴₂

𝑉 = 𝐴₂

𝐴_𝑓𝑟,2𝐿₂ = 𝐴₂Τ𝐿₂

𝐴_𝑓𝑟,2 = 4𝐴_{𝑚𝑖𝑛,2}Τ𝐷_ℎ,2

𝐴_𝑓𝑟,2 = 4𝜎₂

𝐷_ℎ,2 = 𝑏₂𝛽₂ 𝑏₁ + 𝑏₂ + 2𝑎

𝑏

₁

𝑏

₂

𝑎𝑟𝑒 𝑡ℎ𝑒 𝑝𝑙𝑎𝑡𝑒 𝑠𝑝𝑎𝑐𝑖𝑛𝑔 𝑓𝑜𝑟 𝑠𝑖𝑑𝑒 1 𝑎𝑛𝑑 𝑠𝑖𝑑𝑒 2

𝑁

_𝑝

𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑎𝑠𝑠𝑒𝑠

For side 2

1

𝑈₁𝐴₁ = 1

ℎ₁𝐴₁𝜂₀₁ + 𝑅_𝑤 + 1 ℎ₂𝐴₂𝜂₀₂

𝜂₀₁ = 1 − 𝐴_𝑓1

𝐴₁ (1 − 𝜂_𝑓1) 𝜂₀₂ = 1 − 𝐴_𝑓2

𝐴₂ (1 − 𝜂_𝑓2)

𝜂_𝑓 = tanh(𝑀𝑙_𝑓) 𝑀𝑙_𝑓

𝑀 = 2ℎ 𝑘_𝑓𝛿_𝑓

𝑙_𝑓 is the length of the fin 𝛿_𝑓 is the fin thickness

Overall heat transfer for plate-fin heat exchanger

𝐴_𝑤 = 𝐿₁𝐿₂ ∗ 2(𝑁_𝑝 + 1) 𝑅_𝑤 = 𝑎

𝑘_𝑤𝐴_𝑤 Conduction resistance

a) Plain triangular fin

b) Plain rectangular fin c) Wavy fin d) Offset strip fin e) multi-louver fin f) Perforated fin

Some types of plate fin Compact HX

Fin types for plate fin compact heat exchanger

4-Examples

U_=10 m/s

Find h and Dp

Air at p=1 atm T=400 K

Example 10.1

Example 10.1

U_=20 m/s

Find h and Dp

Air at p=2 atm T=500 K

Example 10.2

Example 10.2

m=1500 kg/s A_fr=0.25 m²

Find h and Dp

Air at p=1 atm T=30 C

To=100 C

Example 10.3

Example 10.3