# Heat Transfer and Heat Exchangers - The George W. Woodruff School by 5w73VgGG

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```									                   HEAT TRANSFER AND HEAT EXCHANGERS

INTRODUCTION

The engineer is frequently called upon to transfer energy from one fluid to another. The
transfer is most often effected by a heat exchanger. A steam power plant, for example, includes
several major heat exchangers: boiler, superheater, economizer, and condenser. It also includes
minor heat exchangers such as the gland exhaust condenser and the lube oil cooler. The
principal heat exchangers in an air conditioning system are the evaporator and the condenser.
Other applications abound throughout the mechanical industry.

THEORETICAL CONSIDERATIONS AND THE HEAT TRANSFER PROBLEM

The typical heat transfer problem involves energy transfer from a fluid stream at an
assumed average temperature, through a series of “resistances”, to a destination fluid stream also
at an assumed average temperature. For transfer through a pipe wall, the resistances consist of,
in order, a thermal boundary layer (TBL), a scale or soot layer, the pipe wall, another scale layer,
and the destination TBL.

Each TBL is characterized by a convection (or film) coefficient designated h. Newton’s
Law of Cooling models heat transfer through the TBL:

q   hT  Ts 

where: q " = heat flux or rate of heat transfer per unit area  Btu / h  ft 2 
h = convection (film) coefficient  Btu / h  ft 2  0 R 
T = average temperature of the fluid stream
Ts = surface temperature of the pipe

Fourier’s Law governs transfer by conduction through the pipe wall:

dT
q   k
dr

where: k = thermal conductivity ( Btu / ft  h  0 R )

The negative sign results from the decreasing temperature gradient. Integration yields:

1
2kLT1  T2 
q
ln r2 r1 

where: q = heat transfer rate  Btu / h 
L = length of pipe through which heat transfer occurs  ft 

An experimentally determined fouling factor R''f  ft 2  h  0 R / Btu  accounts for scale resistance.1

THE OVERALL HEAT TRANSFER COEFFICIENT

Heat transfer through the various layers can be represented by an overall heat transfer
coefficient, U, such that:

where: A = heat transfer surface area based on either the inside or outside pipe surface
U = overall heat transfer coefficient (Btu/h-ft2-R)
D = log mean temperature difference, derived below

In analogy with electrical resistance, we can express the Newton and Fourier laws as:

T1  T2                                                       T1  T2
q conv                                                       q cond 
Rconv                                                         Rcond

where:

1                                                          ln r2 r1 
Rconv                                                    Rcond 
hA                                                            2kL

We can then write:

T1  T 2                  T1  T 2
q              
Rtotal        1           ln r2 r1  Rf 2
R f1                     1
                       
A1 h1   A1     2kL       A2      A2 h2

Selecting either the inside or outside area as a basis, factoring out 1/A1 (for example), and letting
A1 = 2r1L, we have:

1
See, e.g., Incropera, p 585, for representative fouling factors.

2
q  AUD

1
1         r ln  r2 r1  r1       r 
U    R1  1
f                    R2  1 
f
 h1             k        r2     r2 h2 

LOG MEAN TEMPERATURE DIFFERENCE

What temperature difference is to be used for D? Because the temperature varies on each
side of the heat exchanger as heat transfer occurs, we must derive a sort of average temperature
difference. Consider a counterflow heat exchanger, which we represent as shown in the figure
below:

Let:    T = temperature of the hot fluid
t = temperature of the cold fluid
h = hot fluid
c= cold fluid
C= heat capacity rate  mc
i = entrance condition
o = outlet condition
D = log mean temperature difference

Ch                                             T               dq                T+dT

t+dt                                 t                          Cc

Ti

 T1                                     dT

dq                                    To
to
dt
 T2

ti

x

q  qc  q h   q h  C h To  Ti                q c  C c t o  t i           dq  C h dT  C c dt  UTdA

3
 dq dq            1   1 
d T   dT  dt  
C        UTdA       
 h   Cc 

C
 h   Cc 


d T        1   1 
2                        2

 T
1
 U 
C       dA

 h Cc  1

 T       T  Ti t o  t i         UA
ln  2   UA o
 T       q                      
 q  To  ti   Ti  t o   UA T2  T1 
   1        h       qc                                              q

T2  T1
D
 T 
ln  2 
 T 
 1

For multipass and cross-flow heat exchangers, the expression for D is modified by a
multiple F, which depends upon heat exchange geometry, number of tube passes and whether the
fluid is mixed (flow transverse to the fluid direction is not prevented) or unmixed (transverse
flow ix prevented by fins or separate tubes). The F-factor can be found in the technical
literature.2

EXTENDED SURFACES (FINS)

Extended surfaces, or fins, are frequently used to increase the heat transfer rate. Fins can
be justified economically when fin effectiveness, defined by the below approximate equation, is

1
 kP  2
 hA   2
f     
 c

where: k = thermal conductivity
P = perimeter of fin
h = convection coefficient
Ac = cross sectional area of fin

Because the temperature varies from the base of the fin to its tip, an adjustment is made to the
total heat transfer surface area, which consists of the uncovered base area plus the fin area.

2
See, e.g., , Incropera and DeWitt, 4 ed, pp. 592-94.

4
Multiplying the total area, At, by an overall surface efficiency, o,3 which is less than unity,
gives:

o 
qt

qt
 1
NA f
1   f   
qmax hAt b       At

where: N = number of fins
Af = area of a single fin
At = total area including base
f = fin efficiency
qt = total heat transfer rate from At
h = convection coefficient
 = Tb - T
Tb = base temperature
T = bulk temperature

Fin efficiency is a function of fin geometry, convection coefficient and thermal
conductivity. Values of fin efficiency for various shapes can be found in the technical literature. 4

Having found both fin and overall surface efficiency, the overall heat transfer coefficient
is modified by multiplying each area (fin plus uncovered base) by the overall surface efficiency.
This step results in the following relation for U:

1
 1    R r ln  r2 r1  r1 R2      r1 
U       f1  1                f
           
o1h1 o1       k         r2 o 2 o 2 r2 h2 

From the definition of fin effectiveness, it is deduced that fins with a high thermal
conductivity, low ratio of perimeter to cross-sectional area and small convection coefficient are
desirable. Because a small convection coefficient improves fin performance, fins are frequently
placed on the gas side of a gas-to-liquid heat exchanger. Automobile radiators and air
conditioning evaporators and condensers are common examples. Fins are often placed on
economizers in boiler designs.

THE CONVECTION HEAT TRANSFER COEFFICIENT

3
See Incropera, Section 3.6.5.
4
See, e.g., Incropera, p 123

5
(FILM COEFFICIENT)

Convection coefficients are determined from empirical correlations found in the
literature. Incropera, for example, includes summaries of convection coefficients for external
and internal flow in the back of chapters 7 and 8. The classical correlation for internal flow is the
Colburn equation:

NuD  0.023Re 4 / 5 Pr1/ 3
D

hD
where: NuD             (the Nusselt number)       k = thermal conductivity of fluid
k
VD
Re           (the Reynolds number)          = kinematic viscosity

                                           k
Pr           (the Prandtl number)                    (thermal diffusivity)
                                          cp

A cautionary note. The engineer must exercise care in choosing and using a correlation.
Particular attention must be given to limits of applicability. Is the flow fully developed?
Laminar or turbulent? What temperature should be used to find properties? What are the
applicable ranges for the Reynolds and Prandtl numbers and the length to diameter ratio?

DESIGN PROCEDURE

   Select an appropriate heat exchanger type.

   Select appropriate convection coefficients.

   Determine needed thermal properties.

   Solve the fundamental equations. Iteration may be required.

qh  mc p T

qc  mc p T

q  UAFD

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