heating water

i29 formulas
⎛
Formula
q
3-1
= A
k ⎞
⎝
⎜can ∆ x ⎠
⎟ (∆T therefore ) be
modified into Formula 3-2:
Formula 3-2:
q = A R (∆T )
Where:
q = rate of heat transfer through the
material (Btu/hr)
R = ∆ x
R = thermal i29 resistance formulas
k (or “R-value”)
of a material (ºF•hr•ft 2 /Btu)
∆T = temperature difference across
the material (°F) ⎛
A = area heat q q = flows AhA(∆T k ⎞
across (ft 2 )
⎝
⎜
∆ x ⎠
⎟ ) (∆T )
The R-value of a material can be
determined based on its thermal
conductivity (k), and its thickness
(∆X). The relationship q = is given as
Formula 3-3. q = sAF A T 4 4
− T
R (∆T 12 ( )
1 2 )
⎡ 1
+ 1 ⎤
⎢ −1⎥
Formula 3-3: ⎣ e 1
e 2 ⎦
R = ∆ x
k
Where:
Re# = vdD
k = thermal conductivity µ of the
material (Btu/°F•hr•ft)
q = hA(∆T )
∆x = thickness of the material in the
direction of heat flow (ft) ⎛ 1 ft ⎞
d = (0.811 in)
The U-values and R-values
⎝
⎜
12 in⎠
⎟
of
materials used to build heat
exchangers q are = sAF critically T 4 4
12 ( − T
1 important.
2 )
In general, materials ⎡ 1and thicknesses
that provide high ⎛ U-value, and thus
low R-value, v allow = 0.408 + 1 ⎤
⎢ ⎞
−1
⎝
⎜ for higher thermal
d 2 ⎠
⎟ f
⎥
⎣ e 1
e 2 ⎦
performance when heat needs
to pass through a solid material
separating the two fluids.
Re# = vdD
CONVECTION
⎛
v = 0.408 µ ⎞
⎛
Convection heat ⎝
⎜
transfer d 2 ⎠
⎟ f occurs = 0.408
⎜
⎝
⎜ 0.811 as
the result of fluid movement. The fluid
⎛ 1 ft ⎞
can be a liquid d or = a (0.811 gas. in)
⎝
⎜
12 in⎠
⎟
When heated, fluids expand. This
lowers their density relative to
surrounding cooler fluid. Lowered
density increases ⎛ buoyancy, which
causes the v warmer = 0.408 ⎞
fluid to rise.
Examples of the
⎝
⎜
latter
d 2
include
⎠
⎟ f
warm
air rising toward the ceiling in a room
and heated water rising to the upper
portion of tank-type water heater.
Both processes occur without
circulators or blowers. As such, they
are examples of “natural” convection.
Convection heat transfer is also
responsible for moving heat between
a fluid and a solid.
For example, consider water at
100ºF flowing along a solid surface
that has a temperature of 120ºF. The
cooler water molecules contacting
the warmer surface absorb heat from
that surface. These molecules are
churned about as the water moves
along. Molecules that have absorbed
heat from the surface are constantly
being swept away from that surface
into the bulk of the water stream and
replaced by cooler molecules. One
can envision this form of convective
heat transfer as heat being “scrubbed
off” the surface by the flowing water.
The speed of the fluid moving over
the surface greatly affects the rate of
convective heat transfer.
Most people have experienced “wind
= 0.06758 chill” ftas cold outside air blows past
( ) 2
⎞
⎟
⎠
⎟
Figure 3-2
5 = 3.1
ft
sec
= 0.06758 ft
them. They feel “colder” compared
to how they would feel if standing
in still air at the same temperature.
The faster the air blows past them,
the greater the rate of convective
heat transfer between their skin or
clothing surfaces and the air stream.
Although the person may feel as if the
moving air is colder, it isn’t. Instead,
they’re experiencing an increased
rate of heat loss due to enhanced
convective heat transfer. Achieving
the same cooling sensation from
calm air would require a much lower
air temperature.
Convective heat transfer increases
with increasing fluid speed. This
happens because a layer of fluid
called the “boundary layer,” which
clings to surfaces, gets thinner as
the fluid’s velocity increases (see
Figure 3-2).
The thinner the boundary layer,
the lower the thermal resistance
between the core of the fluid stream
and the surface. Less thermal
resistance allows for higher rates
of heat transfer between the fluid
molecules in the core of the stream
and the tubing wall.
tube wall
boundary layer of slow moving fluid limits convection
velocity profile of fluid
Note low velocity near tube wall.
"core" of flow stream
heat flows from water,
through boundary layer, tube
wall, into surrounding material
28
⎛
v = 0.408 ⎞
⎛
⎝
⎜
d 2 ⎠
⎟ f = 0.408
⎞
ft
⎜ ⎟ 5 = 3.1
⎝
⎜ 0.811 ⎠
⎟ sec
( ) 2