Chapter12
Heat transfer to fluids
without phase change
Regimes of heat transfer in fluids

A fluid being heated or cooled may be
flowing in different flow patterns. Also,
the fluid may be flowing in forced or
natural convection.
At ordinary velocities the heat generated
from fluid friction is negligible in
comparison with the heat transferred
between the fluids.

Because the situations of flow at the
entrance to a tube differs from those well
downstream from the entrance, the
velocity field and associated temperature
field may depend on the distance from the
tube entrance
The properties of the fluid-viscosity,
thermal conductivity, specific heat, and
density are important parameters in heat
transfer.
Each of these, especially viscosity, is
temperature-dependent.
Heat transfer by forced convection
in turbulent flow

Perhaps the most important situation
in heat transfer is the heat flow in a
stream of fluid in turbulent flow.
Since the rate of heat transfer is greater in
turbulent flow than in laminar flow, most
equipment is operated in the turbulent
range.
A dimensional analysis of the heat flow
to a fluid in turbulent flow through a
straight pipe yields dimensionless
relations.
 du  c p  
hd
 f
,

k
 
 
(12-27)
The three groups in Eq(12-27) are
recognized as the Nusselt(Nu),
Reynolds(Re), and Prandtl (Pr) numbers
respectively.
The Nusselt number for heat transfer from a
fluid to a pipe or from a pipe to a fluid
equals the film coefficient multiplied by d/k
hd
Nu 
k
The film coefficient h is the average value
over the length of the pipe
Prandtl number Pr is the ratio of the
diffusivity of momentum μ/ρ to the
thermal diffusivity k/ρcp
Pr 
cp 
k
The Prandtl number of a gas is usually
close to 1（0.69 for air, 1.06 for steam). The
Prandtl number of gases is almost
independent of temperature because the
viscosity and thermal conductivity both
increase with temperature at about the
same rate.
Empirical equation
For heat transfer to and from fluids that follow
the power-law relation, the dimensionless
relation becomes
hd
du  m c p  n
 c(
) (
)
k

k
To use the dimensionless relation, the constant
c and index m, n must be known.
A recognized empirical correlation, for
long tubes with sharp-edged entrances,
is the Dittus-Boelter equation
hi d
0.8
n
Nu 
 0.023 Re Pr
k
Where n is 0.4 when the fluid is being
heated and 0.3 when it is being cooled.
A better relationship for turbulent flow is
known as the Sieder-Tate equation
Nu  0.023Re
0.8
 0.14
Pr (
)
(12-32)
w
1/ 3
Equation(12-32) should not be used for
Reynolds numbers below 6000 or for
molten metals, which have abnormally
low Prandtl number.
Effect of tube length

Near the tube entrance, where the
temperature gradients are still forming,
the local coefficient hx is greater than h
for fully developed flow.
In entrance, hx is quite large, but hx value
drops rapidly toward h in a comparatively
short length of tube.
Average value of hi in turbulent flow. Since
the temperature of the fluid changes from
one end of the tube to the other and fluid
properties µ , cp and k are all function of
temperature, the local value of hi also varies
from point to point along the tube.

The relation of local heat transfer coefficient
hi and long tube h  is as follows
D 0.7
hi / h  1  ( )
L

When L approaches infinite, hi is close to the
h  of long tube.


For laminar flow, the relation of Nu and Pr
and Re is
Nu  A(Re Pr)1/ 3
(12.25)

For gases
the effect of temperature on hi is small.
The increase in conductivity and heat
capacity with temperature offset the rise
in viscosity, giving a slight increase in hi.

For liquids
the effect of temperature is much greater
than for gases because of the rapid
decrease in viscosity with rising
temperature.
The effects of k, cp, and µ in Eq(12-36) all
act in the same direction, but the increase
in hi with temperature is due mainly to the
effect of temperature on viscosity.
In practice, an average value of hi is
calculated and used as a constant in
calculating the overall coefficient U.
the average value of hi is computed by
evaluating the fluid properties k, cp, and µ
at average fluid temperature, defined as
the arithmetic mean between the inlet and
outlet temperatures.
Estimation of wall temperature tw
The estimation of tw requires an iterative
calculation based on the resistance
equation
tm To tw
ti



1
1
b do 1 do
U o h o k d m h i di
To determine tw the wall resistance can
usually be neglected
t m
 ti

1
1 do
Uo
h i di
Substituting Uo, gives
1
hi
(12-38)
ti 
tm
1 1 di

hi ho d o
Cross sections other than circular
To use Eq(12-30) for cross section other
than circular it is only necessary to
replace the diameter in both Reynolds and
Nusselt number by the equivalent
diameter de.
de is defined as 4 times the
hydraulic radius rH. The method is
the same as that used in
calculating friction loss.
Heat transfer in transition region
between laminar and turbulent flow
Equation (12-32) applies only for Reynolds
numbers greater than 6000.
The range of Reynolds numbers between
2100 and 6000 is called the transition
region, and no simple equation applies
here.
A graphical method therefore is used.
The method is based on a common plot of
the Colburn j factor versus Re, with lines
of constant value of L/D
The heat transfer coefficient can be
calculated by following equation
1
3
d
  
Nu  2  Re Pr   
 4L
  w 
0.14
Heating and cooling of fluids in
forced convection outside tubes
The mechanism of heat flow in forced
convection outside tubes differs from that
of flow inside tubes.
The local value of heat-transfer coefficient
varies from point to point around
circumference in forced convection outside
tube.
In Fig12.5, the local value of the Nusselt
number is plotted radially for all points
around circumference of the tube.
Nuθis maximum at the front and back of the
tube and a minimum at the sides.
In practice, the variations in the local
coefficient are often no importance, and
average values based on the entire
circumference are used.
fluids flowing normal to a
single tube
The variables affecting the coefficient of
heat transfer to a fluid in forced
convection outside a tube are Do, the
outside diameter of the tube; cp, μ, and k,
the specific heat, the viscosity, and
thermal conductivity, respectively, of the
fluid; and G, the mass velocity.
Dimensional analysis gives
 Do G c p  
ho Do
 f
,

k
k 
 
Nusselt number is only a function of the
Reynolds number.
The experimental data for air are plotted in this
way in Fig12.6
For heating and cooling liquids flowing
normal to single cylinders the following
equation is used
ho Do  c p  f

kf  kf



0.3
 DoG 
 0.35  0.56 
  
 f 
0.52
Natural convection
Consider a hot, vertical plate in contact with
the air in a room.
The density of the heated air immediately
adjacent to the plate is less than that of the
unheated air at a distance from the plate,
and the buoyancy of the hot air causes an
unbalance between the vertical layers of air
of differing density.
Temperature difference between the surface
of plate and the air causes a heat transfer.
Natural convection in liquid follows the same
pattern. The buoyancy of heated liquid layers
near a hot surface generates convection
currents just as in gases.
For single horizontal cylinders, the heat
transfer coefficient can be correlated by
equation containing three dimensionless
groups
Nu=f(Pr, Gr)
Gr: Grashof number
Pr: Prandtl number
3 2


c

D
hDo
p f
o  f  g t
f
,

2
 k

k

f
 f

(12-67)
The coefficient of thermal expansion β is a
property of fluid
Fig 12.8 shows a relationship, which
satisfactorily correlates experimental data
for heat transfer from a single horizontal
cylinder to liquids or gases
For magnitudes of log Gr Pr of 4 or more, the line
of Fig 12.8 follows closely the empirical equation
Nu  0.53  Gr Pr  f
0.25
Natural convection to air from vertical
shapes and horizontal plates
Equations for heat transfer in natural
convectionbetween fluids and solids of definite
geometric shape are of the form
n
 c p  f L   g t 
hL
 b

2

(12-73)
k
f
 kf

Values of the constants b and n for various
conditions are given in Table 12.4
3
2
f




A double pipe heat exchanger is used to condense
the saturated toluene vapor (2000kg/h) into
saturated liquid. The condensation temperature and
latent heat of toluene are 110 oC and 363kJ/kg,
respectively. The cold water at 20 oC (inlet
temperature) and 5000kg/h goes through the pipe
(di=50 mm) fully turbulently. If the individual heat
transfer coefficient hi of water side is 2100 w/(m2 K),
and heat resistances of pipe wall as well as toluene
side are much larger than that of water side (this
means both resistances can be ignored), find:
Outlet temperature of cold water, in oC.
Pipe length of exchanger.
In order for mass flow rate of toluene to be double, if
the mass flow rate of cold water at the same inlet
temperature (20 oC) is double, what is the pipe
length of new exchanger to be required?















Solution: Heat balance q=m1=m2Cp(Tcb-Tca)
2000363=50004.19(Tcb-20)
(1)Outlet temperature of cold water Tcb=54.65oC
(2)U=h (from the problem)
∆T1=110-54.65=55.35, ∆T2=110-20=90
∆T=(∆T1+∆T2)/2=72.68 (since ∆T2/∆T1<2)
L=q/(Ud∆T)=20003631000/3600/(21000.0572.68)=8.42m
(3) q’=2qm1=2m2Cp(T’cb-Tca)
Outlet temperature of cold water Tcb=54.65oC
∆T’=(∆T1+∆T2)/2=72.68
Fully developed turbulent flow, hRe0.8~m0.8~u0.8
h’/h=20.8 , h’=1.74h
q’=1.74hdL’ ∆T’= 2m1
q=hdL ∆T= m1
L’/L=2/1.74 so L’=28.42/1.74=9.68m


A single pass (1-1) shell-tube exchanger is made of
many 252.5 mm tubes. Organic solution,
u=0.5m/s, m(mass flow rate)= 15000kg/h, Cp=1.76
kJ/kg. oC, =858 kg/m3, passes through the tube.
The temperature changes from 20 to 50 oC. The
saturated vapor at 130 oC condenses to the
saturated water, which goes through the shell. The
individual heat transfer coefficients hi and ho in the
pipe and shell are 700 and is 10000 W/m2 oC,
respectively. The thermal conductivity k of pipe wall
is 45 W/m. oC. If the heat loss and resistances of
fouling can be ignored, find(1)Overall heat transfer
coefficient Uo.（based on outside tube area）and
LMTD.
(2) Heat transfer area, number of pipes and length of
pipes.