SENSIBLE HEAT FLUX
By R.L. Snyder and K.T. Paw U
Copyright  Regents of the University of California
Created  June 28, 2000
Air molecules move
about at near the speed of sound (300 m s^{1}), and they are
constantly colliding with their surroundings and each other. In general, if a molecule travels from some
height z to height z + l, where it collides with another
molecule, the distance l is called
the “mean free path”. After colliding with the new molecule, the energy (E_{k}) is transferred to the new
molecule and the temperature increases to that of the original air molecule.
_{} Joules
where m is the mass of the molecule (kg) and C_{p} (J kg^{1}K^{1})
is the specific heat per unit mass at constant pressure. Using a Taylor’s approximation, we can write
_{} Kelvin
and, by substitution,
we get
_{} Joules
To know the heat
transfer per unit area, we need to know the number of active molecules moving with
a vertical velocity component at any given time. If we let _{} equal the number of
molecules per unit volume, then, at any given time, _{} is the number with a
vertical velocity. The flux (F) of molecules crossing a horizontal
plane per unit time is the product of _{} and the average
vertical velocity (c) of the active
molecules.
_{}
_{}
The heat flux density equals the product of the molecular flux density and the heat transferred by each molecule.
_{} W
m^{2}
However, the product
of the number of active molecules and the mass per molecule (Nm) is the mass of air per unit volume
or density (_{}), so by rearranging terms, we get a formula for sensible
heat flux density (H)
_{} W
m^{2}
Here, _{} is the “thermal
diffusivity” in m^{2} s^{1}.
A typical value for air near Earth’s surface is k = 2.2 ´ 10^{3}
m^{2} s^{1}.
Because of sensor
limitations, it is not possible to measure temperature differences over small
distances. Consequently, we measure the
temperature at two heights and assume that the gradient is constant between the
two levels. Then we use the following
equation
_{} W
m^{‑2}
where z_{2} is further from the
surface than z_{1}. Also, we
usually don’t know the value for k,
so we use the substitution
_{} m
s^{1}
to get H in terms of a conductance (g_{h}) or resistance (r_{h}) to sensible heat
transfer.
_{} W
m^{‑2}
The conductance (g_{h}) is the rate at which one
cubic meter of heat passes through one square meter of surface area. Conductance has the units m s^{1}
and resistance has units s m^{1}.
Temperature gradients
are often large near a surface but become smaller with distance from the
surface. Therefore, the temperature
approaches a fixed value _{} if the distance is
far relative to the size of the object.
The zone between the surface and where the temperature reaches _{} is called the
boundary layer for sensible heat flux. One could estimate H using _{} and the surface temperature
(T) as
_{} W
m^{‑2}
where d is the “boundary
layer height” or distance from the surface to where the temperature is near _{}. However, d is difficult to
measure, so we assume that d
is related to the ratio of the object dimension d to the boundary layer height (d). We
then define a dimensionless “Nusselt” number as
_{}
Because 1/d = Nu/d,
we get H as
_{} W
m^{2}
where d is the object dimension. The meaning of d depends on the shape of the object. For a flat plate, d is
length parallel to the flow. For
cylinders that are normal to the flow, d
is the diameter. For spheres, d is the diameter. Values for Nu are determined experimentally.
When air blows around
an object (forced convection), the flow can be turbulent or viscous
(laminar). The wind speed and size and
shape of the object determine if the flow is viscous or turbulent. At low wind speed, the flow is generally
laminar, but it becomes turbulent at higher velocities. The dimensionless
“Reynolds” number (Re) is related to
onset of turbulence.
_{}
where n is the kinematic
viscosity (»1.5
´ 10^{5}
m^{2} s^{1} for air), d
is the object dimension in m, and U
is the wind speed (m s^{1}) relative to the object. R_{e}
represents the ratio of inertial forces (related to acceleration of the fluid)
to viscous forces (due to molecular transport of momentum). At low wind speeds, the inertial and viscous
forces are similar, Re is small and
the flow is orderly. For high Re, inertial forces are much larger than
viscous forces and the airflow becomes turbulent.
The distance from an
object where the temperature approaches _{} depends on whether
the airflow is viscous or turbulent.
Therefore, Nu is a function of
Re.
When air is blowing around an object, Nu can be estimated from Re
as
_{}
where b, c, and n are empirically determined for a given object shape.
From Monteith (1973)
Object 
Re 
Nu 
Flat Plate
w/parallel flow 
<
2 ´ 10^{4} 
0.60
Re^{0.5} 

>
2 ´ 10^{4} 
0.03
Re^{0.8} 
Cylinders w/normal
flow 
10^{1}
 10^{4} 
0.32
+ 0.51 Re^{0.52} 

10^{3}
 5 ´
10^{4} 
0.24
Re^{0.60} 

4
´ 10^{4}
 4 ´
10^{5} 
0.024
Re^{0.81} 
Spheres 
0
– 300 
2
+ 0.54 Re^{0.5} 

50
– 1.5 ´
10^{5} 
0.34
Re^{0.6} 
When the air around an
object is not moving, the object may heat or cool rapidly. This causes a density change near the surface,
which results in free or natural convection of heat. Since there is no wind, there is no Re. Then Nu becomes a function of the “Grashof”
number (Gr).
_{}
where g is the acceleration due to gravity
(9.8 m s^{2}) and a_{t}
is the coefficient of expansion (a_{t}=1/273
for an ideal gas). The Grashof number represents the product of the ratio of
buoyancy to viscous forces and the ratio of inertial to viscous forces. At higher temperatures, Gr is bigger and there is more free convection. For air, Gr is approximately
_{}
Nu is determined from Gr as
_{}
where b and m are determine empirically.
The object dimension d is the smallest
of length or width for horizontal flat plates, length for vertical flat plates,
and diameter for cylinders and spheres.
Object 
Gr 
Nu 
Horizontal Flat
Plate 
<
10^{5} 
0.60
Gr^{0.25} 

>
10^{5} 
0.13
Gr^{0.33} 
Horizontal Cylinder 
10^{4}
– 10^{9} 
0.48
Gr^{0.25} 

>
10^{9} 
0.09
Gr^{0.3} 
Vertical Flat Plate 
10^{4}
– 10^{9} 
0.58
Gr^{0.25} 

10^{9}
– 10^{12} 
0.11
Gr^{0.3} 
Spheres 
<
2 ´10^{10} 
2
+ 0.54 Gr^{0.25} 