1-Basic types of heat exchangers 2-LMTD method

1

King Abdulaiz University Faculty of Engineering

Mechanical Engineering Department

MEP460 Heat Exchanger Design

Sep. 2018

LMTD and Effectiveness-NTU Heat exchanger methods

1-Basic types of heat exchangers 2-LMTD method

3-Effectivness-NTU method 4-Compcat heat exchangers 5-Examples

Ch. 11 Incropera (Heat Exchangers)

1-Basic types of heat exchangers

Classification according to flow arrangements Parallel flow arrangement

Counter flow arrangement Cross flow heat arrangement

Counter flow arrangement Parallel flow arrangement

Cross flow heat arrangement

Both fluids are Unmixed cross flow HX with fins

Mixed unmixed heat exchanger with no fins

Classification of heat exchangers according to flow phase

Liquid to liquid HX Liquid to Gas HX Gas to Gas HX

Example Cooling of oil using sea water Using exhaust gases to heat water

(Car radiator)

Heating of air using exhaust gases

Other very common types of heat exchangers

a) Shell and tube heat exchangers b) Compact heat exchangers

Plate-fin compact heat exchanger Tube-fin compact heat exchanger c) Plate heat exchangers

d) Cooling towers

a) Shell and tube heat exchangers

One shell pass and two tube passes Two shell passes and 4 tube passes

b) Compact heat exchangers

Surface area density= 𝛼 A/V [m²/m³]

For compact heat

exchangers 𝛼

> 700^𝑚2

𝑚³

c) Plate and frame (Gasketed) heat exchangers

c) Plate and frame heat exchangers

d-Cooling Towers

d-Cooling Towers

Overall heat transfer coefficient

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

2-Logarethmic Mean Temperature Difference

LMTD

method

𝑇

_ℎ𝑖

𝑇

_ℎ𝑜

𝑇

_𝑐𝑜

𝑇

_𝑐𝑖

𝑞 = ሶ𝑚_ℎ(ℎ_ℎ𝑖 − ℎ_ℎ𝑜) 𝑞 = ሶ𝑚_ℎ𝐶_𝑝,ℎ (𝑇_ℎ𝑖 − 𝑇_ℎ𝑜) 𝑞 = 𝐶_ℎ(𝑇_ℎ𝑖 − 𝑇_ℎ𝑜)

𝑞 = ሶ𝑚_𝑐(ℎ_𝑐𝑜 − ℎ_𝑐𝑖) 𝑞 = ሶ𝑚_𝑐𝐶_𝑝,𝑐(𝑇_𝑐𝑜 − 𝑇_𝑐𝑜) 𝑞 = 𝐶_𝑐(𝑇_𝑐𝑜 − 𝑇_𝑐𝑖) Hot fluid

Cold fluid

𝐶_ℎ = ሶ𝑚_ℎ𝐶_𝑝,ℎ heat capacity rate [W/K] of the hot fluid 𝐶_𝑐 = ሶ𝑚_𝑐𝐶_𝑝,𝑐 heat capacity rate [W/K] of the cold fluid

Temperature distribution

2-Logarethmic Mean Temperature Difference

LMTD

^method

Heat transfer for a small element of thickness

dx and heat transfer area of dA

Parallel heat exchanger arrangement

Δ𝑇 = 𝑇_ℎ − 𝑇_𝑐 𝑑𝑞 = − ሶ𝑚_ℎ𝐶_𝑝,ℎ𝑑𝑇_ℎ = −𝐶_ℎ𝑑𝑇_ℎ 𝑑𝑞 = ሶ𝑚_𝑐𝐶_𝑝,𝑐𝑑𝑇_𝑐 = 𝐶_𝑐𝑑𝑇_𝑐

𝑑𝑞 = 𝑈𝑑𝐴(𝑇_ℎ − 𝑇_𝑐) 𝑑 Δ𝑇 = 𝑑𝑇_ℎ − 𝑑𝑇_𝑐

𝑑 Δ𝑇 = −𝑑𝑞

𝐶_ℎ − 𝑑𝑞 𝐶_𝑐

2-Logarethmic Mean Temperature Difference

LMTD

method

𝑑 Δ𝑇 = −𝑑𝑞

𝐶_ℎ − 𝑑𝑞 𝐶_𝑐 𝑑 Δ𝑇 = −𝑑𝑞 1

𝐶_ℎ + 1 𝐶_𝑐 𝑑𝑞 = 𝑈𝑑𝐴Δ𝑇 𝑑 Δ𝑇 = −𝑈𝑑𝐴Δ𝑇 1

𝐶_ℎ + 1 𝐶_𝑐 𝑑Δ𝑇

Δ𝑇 = −𝑈𝑑𝐴 1

𝐶_ℎ + 1 𝐶_𝑐

𝑑Δ𝑇

Δ𝑇 = −𝑈𝑑𝐴 𝑇_ℎ𝑖 − 𝑇_ℎ𝑜

𝑞 + 𝑇_𝑐𝑜 − 𝑇_𝑐𝑖 𝑞

𝑞 = 𝐶_ℎ(𝑇_ℎ𝑖 − 𝑇_ℎ𝑜)

𝑞 = 𝐶_𝑐(𝑇_𝑐𝑜 − 𝑇_𝑐𝑖) 𝐶_ℎ = 𝑞

𝑇_ℎ𝑖 − 𝑇_ℎ𝑜

𝐶_𝑐 = 𝑞 𝑇_𝑐𝑜 − 𝑇_𝑐𝑖

𝑑Δ𝑇

Δ𝑇 = −𝑈𝑑𝐴

𝑞 𝑇_ℎ𝑖 − 𝑇_𝑐𝑖 − 𝑇_ℎ𝑜 − 𝑇_𝑐𝑜

ln Δ𝑇₂

Δ𝑇₁ = 𝑈𝐴

𝑞 𝑇_ℎ𝑜 − 𝑇_𝑐𝑜 − 𝑇_ℎ𝑖 − 𝑇_𝑐𝑖

Δ𝑇₂ = 𝑇_ℎ𝑜 − 𝑇_𝑐𝑜 Δ𝑇₁ = (𝑇_ℎ𝑖−𝑇_𝑐𝑜) 𝑞 = 𝑈𝐴 Δ𝑇₂ − Δ𝑇₁

ln(Δ𝑇₂ΤΔ𝑇₁) 𝑞 = 𝑈𝐴Δ𝑇_𝑙𝑚

2-Logarethmic Mean Temperature Difference

LMTD

method

Δ𝑇_𝑙𝑚 = 𝐿𝑀𝑇𝐷

Logarithmic Mean Temperature Difference LMTD

In general

𝑞 = 𝑈𝐴Δ𝑇

_𝑙𝑚

F is the correction factor

F=1 for the following cases:

a) Pure counter current flow b) Pure parallel flow

c) Evaporators (phase change) d) Condensers (phase change)

Δ𝑇

_𝑙𝑚

= Δ𝑇

_{𝑙𝑚,𝐶𝐹}

𝐹

Δ𝑇_𝑙𝑚 is called Logarithmic Mean Temperature

Difference=LMTD

For other geometries, see the next slides

2-Logarethmic Mean Temperature Difference

LMTD

method

Ref.: Cengle heat transfer an engineering approach, 2^nd edition

LMTD correction factor

Ref.: Cengle heat transfer an engineering approach, 2^nd edition

LMTD correction factor

Ref.: Cengle heat transfer an engineering approach, 2^nd edition

LMTD correction factor

Ref.: Cengle heat transfer an engineering approach, 2^nd edition

LMTD correction factor

Special Cases

SIZING PROBLEMS :

• Heat rate q required is known. Outlet temperatures can be calculated.

• Calculate DT

_lm,CF

and obtain the correction factor (F) if necessary

• Calculate the overall heat transfer coefficient, U.

• Determine A using 𝑞 = 𝑈𝐴ΔT

_LM,CF

F

The LMTD method is not as easy to use for performance analysis….

LMTD Method

Example 11.1

Example 11.1

Example 11.1

Example 11.1

Example 11.1

Heat transfer coefficient Correlations used from Ch.8 a-Turbulent flow inside a pipe

b-Laminar flow inside an annulus

𝜖 = 𝑓(𝐶

_𝑟

, 𝑁𝑇𝑈) 𝑁𝑇𝑈 = 𝑓(𝐶

_𝑟

, 𝜖)

𝐶_𝑟 = 𝐶_𝑚𝑖𝑛 𝐶_𝑚𝑎𝑥

NTU =Number of heat transfer units 𝑁𝑇𝑈 = 𝑈𝐴 𝐶_𝑚𝑖𝑛 𝑞_𝑚𝑎𝑥 = 𝐶_𝑚𝑖𝑛Δ𝑇_𝑚𝑎𝑥 = 𝐶_𝑚𝑖𝑛(𝑇_ℎ𝑖 − 𝑇_𝑐𝑖)

𝜖 = 𝑞

𝑞_𝑚𝑎𝑥

𝑞 = 𝑞

_𝑚𝑎𝑥

𝜖

Heat Exchanger effectiveness

𝜖 − 𝑁𝑇𝑈 Method

Define the maximum possible heat

Define the maximum and minimum heat capacity rate

𝐶_𝑚𝑖𝑛 = 𝑚𝐶ሶ _{𝑝 𝑚𝑖𝑛} 𝐶_𝑚𝑎𝑥 = 𝑚𝐶ሶ _{𝑝 𝑚𝑎𝑥}

𝜖 − 𝑁𝑇𝑈 Method

Assume C_h=C_min

𝜖 = 𝑞

𝑞_𝑚𝑎𝑥 = 𝐶_ℎ(𝑇_ℎ𝑖 − 𝑇_ℎ𝑜) 𝐶_ℎ(𝑇_ℎ𝑖 − 𝑇_𝑐𝑖)

Consider counter current heat exchanger

From LMTD analysis

Effectiveness  as a function of C

_r

and NTU

NTU relations as a function of  and C_r for different heat exchangers

𝜖 − 𝑁𝑇𝑈

𝜖 − 𝑁𝑇𝑈

𝜖 − 𝑁𝑇𝑈

Cross flow heat exchanger Both fluid un-mixed

Cross flow heat exchanger One fluid mixed and the other un-mixed

PERFORMANCE ANALYSIS

• Usually inlet temperatures, mass flow rates and heat

exchanger area A, are known. Calculate the capacity ratio C

_r

= C

_min

/C

_max

and NTU = UA/C

_min

from input data

• Determine the effectiveness from the appropriate charts or

-NTU equations for the given heat exchanger and specified flow arrangement.

• When  is known, calculate the total heat transfer rate using 𝜀 =

^𝑞

𝑞_𝑚𝑎𝑥

, where 𝑞

_𝑚𝑎𝑥

= 𝐶

_𝑚𝑖𝑛

Δ𝑇

_𝑚𝑎𝑥

• Calculate the outlet temperature.

𝜖 − 𝑁𝑇𝑈

Example 11.3

Example 11.3

Example 11.3

Example 11.3

Example 11.3

Compact heat exchangers

MEP460

Heat Exchanger Design

Ch. 11 Incropera, 6

^th

Edition

Feb. 2018

Compact heat exchangers

Surface heat transfer area over volume 𝛼

Tube fin compact heat exchangers

Tube fin compact heat exchangers

Plat fin compact heat exchangers

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

Chapter 8 can be used to find the heat transfer coefficient inside pipes or ducts

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

_ff

Fin area/total area=A

_f

/A

_o

𝜎 = 𝐹𝑟𝑒𝑒 𝑓𝑙𝑜𝑤 𝑎𝑟𝑒𝑎

𝐹𝑟𝑜𝑛𝑡𝑎𝑙 𝑎𝑟𝑒𝑎 = 𝐴_𝑓𝑓 𝐴_𝑓𝑟

𝛼 = Surface area/Volume

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

Friction factor f

_{Δ𝑃 = 𝑓} ^𝐿

𝐷 𝜌 𝑉² 2

𝑅𝑒 = 𝐺𝐷

_ℎ

/𝜇

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

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

Circular fin on circular tubes

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

Continuous fin on circular tubes

𝐹𝑖𝑛 𝑎𝑟𝑒𝑎

𝑇𝑜𝑡𝑎𝑙 𝑎𝑟𝑒𝑎 = 𝐴

_𝑓

𝐴

𝛼 = 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

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

Example 11.6 Continue

Example 11.6 Continue

Example 11.6 Continue

𝑅𝑒 = 𝐺𝐷_ℎ 𝜇

Example 11.6 Continue

Example 11.6 Continue

Example 11.6 Continue