Heat Transfer theory

Case (i) Heat transfer w/o shield.

Qw / o = σ A(T14 − T24 )
1 + 1 −1
ε1 ε2

Case (iii) Heat transfer with shield.

Steady Q13 = Q32

σ A(T14 − T34 ) = σ A(T34 − T24 )
1 + 1 −1 1 + 1 −1
ε1 ε3 ε3 ε2

Special case let ε1 = ε2 = ε3 = ε

( ) ( )T14 − T34 = T34 − T24

T34 = T14 + T24
2

Qwith Shield = Q13 = Q32

= σ A(T14 − T34 )
1+1 −1
ε1 ε2

σ A(T14 − ⎛ T14 + T24 ⎞
⎜ 2 ⎟
⎝ ⎠
Qwith shield =
1 −1
ε

( )Qwith
shield = σ A T14 − T24

2 ⎛ 2 − 1⎠⎞⎟
⎝⎜ ε

( )Qwith
shield = σA T14 − T24
2 −1
ε

Qwith shield = 1 = 1
Qwithout shield 2 n +1

n = number of shields
* Radiative heat exhcnage between two non-black bodies :

Radiation emitted by (1) = ε1σ A1T14
emitted by (1) = ε1α2σ A1F12T14
absorbed by (1) = ε 2α1σ A2 F21T24

Net exchange
Q12 = ε1α 2σ A1F12T14 − ε 2α1σ A2 F21T24

A1F12 = A2 F21
α = ε so thermal equition

Q12 = ε1ε 2σ A1F12 ⎣⎡T14 − T24 ⎤⎦
ε1ε2 = euivalent emissivity

5. Heat Exchanger

Classification
I. Based on contact
a) Direct Contact
b) Indirect Contact
ll. Based on direction of flow of fluid
a) Parallel flow
b) Counter flow
c) Cross flow

Design of Heat exchangers

I. LMTD ( Logarithmic mean temperature difference method) - This method is used
when outlet temp. of both fluids are known. With this method the surface area of heat
exchanger can be calculated.

II. Effectiveness NTU method – This method is used when the outlet temperature are
unknown.

Logarithmic mean temperature difference (LMTD)

To begin with, we take U to be a constant value. This is fairly reasonable in compact single-
phase heat exchangers. In larger exchangers, particularly in shell-and-tube configurations and
large condensers, U is apt to vary with position in the exchanger and/or with local
temperature. But in situations in which U is fairly constant, we can deal with the varying
temperatures of the fluid streams by writing the overall heat transfer in terms of a mean
temperature difference between the two fluid streams:

The determination of ∆Tmean for such arrangements proceeds as follows: the differential
heat transfer within either arrangement is

where the subscripts h and c denote the hot and cold streams, respectively; the upper and
lower signs are for the parallel and counterflow cases, respectively; and dT denotes a change
from left to right in the exchanger

After lengthy manipulation we get for either parallel or counterflow,

Effectiveness-NTU method

Often we begin with information such as is shown in Fig. 5-1. If we sought to calculate Q in
such a case, we would have to do so by guessing an exit temperature

Such problems can be greatly simplified with the help of the so-called effectiveness-NTU
method. This method was first developed in full detail by Kays and London in 1955, in a
book titled Compact Heat Exchangers. We should take particular note of the title. It is with
compact heat exchangers that the present method can reasonably be used, since the overall
heat transfer coefficient is far more likely to remain fairly uniform. The heat exchanger
effectiveness is defined as

whereCmin is the smaller of Cc and Ch. The effectiveness can be interpreted as

A second definition that we will need was originally made by E.K.W. NusseltThis is the
number of transfer units (NTU):

Effectivenessequation for the parallel single-pass heat exchanger:
Effectiveness equation for the counter flow single-pass heat exchanger:

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