Page 266 - Fluid mechanics, heat transfer, and mass transfer

CONVECTIVE HEAT TRANSFER 247

For air, these equations can be approximated respec-
tively by
0:25
h c ¼ 1:3ðDT=LÞ ð9:7Þ
and

0:25
h c ¼ 1:8ðDTÞ : ð9:8Þ
FIGURE 9.2 Hydrodynamic and thermal boundary layers. ➢ The equations for air are not dimensionless where
2

DT is in C, L or D in m, and h c in J/(m s C). The

& If the Prandtl number is less than 1 (low Prandtl characteristic dimension to be used in the calcu-
number fluids), which is the case for air and many lation of N Nu and N Gr in these equations is the
other gases at standard conditions, the thermal height of the plane or cylinder.
boundary layer is thicker than the hydrodynamic & Natural convection about horizontal cylinders such
boundary layer. Mercury and liquid metals have far as a steam pipe:
less values of Prandtl numbers than gases and bound-
ary layer thicknesses are much higher. 0:25
N Nu ¼ 0:54ðN Pr N Gr Þ for laminar flow in
& When Prandtl number ffi 1, the boundary layer 3 9
range 10 < N Pr N Gr < 10 : ð9:9Þ
thicknesses are nearly same.
➢ For air the approximate equations are the same as
. ‘‘Prandtl number relates the thickness of the hydrody-
for vertical surfaces, except for higher ranges of
namic boundary layer to the thickness of the thermal
N Pr N Gr ,theexponentforDTis0.33insteadof0.25.
boundary layer.’’ True/False?
& For horizontal planes, the equations for cylinders
& True.
may be used, using L in place of D of the cylinder
. What are the dimensionless groups that are significant
wherever D is used in N Nu and N Gr .
in natural convection heat transfer?
& In the case of horizontal planes, cooled when facing
& Grashof number.
upward, or heated when facing downward, which
& Rayleigh number.
appear to be working against natural convection
. Define Grashof number. For what type of heat transfer,
circulation, half of the value of h c , found in the given
this number is of significance?
correlations, may be used.
& Grashof number is given by N Gr ,
. What is Colburn J H factor? What is its significance?
3 2
2
L r bg Dt=m ¼ ratio of buoyancy force to Give the appropriate equations.
viscous force acting on the fluid: & J-factor is evolved in observing analogous conditions
between momentum and heat transfer.
ð9:3Þ
& The analogy between heat and momentum is evolved
& Natural convection heat transfer.
by assuming that diffusion of momentum and heat
. Define Rayleigh number.
occurs by essentially the same mechanism so that a
& Rayleigh number is the product of Grashof and
relatively simple relationship exists between the
Prandtl numbers.
momentum and heat diffusion coefficients.
3 2
N Ra ¼ N Gr N Pr ¼ L r gbC p DT=ma: ð9:4Þ & Colburn has defined J-factor for momentum as
& It governs natural convection heat transfer.
J M ¼ f=2; ð9:10Þ
. What are the equations that are used in correlating
natural convection heat transfer? where f is the friction factor. J-factor for heat transfer
& Natural convection about vertical cylinders and was assumed to be equal to J-factor for momentum
planes: transfer and designated as J H .
& As per Colburn’s analogy,
0:33 4 9
N Nu ¼ 0:53 ðN Pr N Gr Þ ; for 10 < N Pr N Gr < 10
ð9:5Þ 2=3
J M ¼ f=2 ¼ J H ¼ h=C p GðN Pr Þ ; ð9:11Þ
and
where h is the heat transfer coefficient, G is the mass
0:33 9 12
N Nu ¼ 0:12 ðN Pr N Gr Þ for 10 < N Pr N Gr < 10 : velocity ¼ Vr, velocity multiplied by density of the
fluid.
ð9:6Þ