heating water

R = ∆ x
k
The rate of heat exchange by thermal radiation between
two parallel flat q = surfaces hA(∆T having ) the same area can be
estimated using Formula 3-5.
Formula 3-5:
( )
q = sAF T 4 4
− T
12 1 2
⎡ 1
+ 1 ⎤
⎢ −1⎥
⎣ e 1
e 2 ⎦
Where:
q = rate of heat transfer from hotter to cooler surface by
thermal radiation Re# (Btu/hr) = vdD
s = Stefan Boltzmann constant
µ
= 0.1714x10 -8 Btu/
hr•ft 2 •ºR 4
F 12 = shape factor between the two surfaces (unitless)
A = area of either surface (ft 2 )
⎛ 1 ft ⎞
T 1 = absolute temperature d = (0.811 of in) the hotter surface (ºR)
T 2 = absolute temperature of the ⎝
⎜
cooler 12 in⎠
⎟ = 0.06758 ft
surface (ºR)
e 1 = emissivity of the hotter surface (unitless)
e 2 = emissivity of the cooler surface (unitless)
It’s also important to understand that the temperatures (T1)
and (T2) in Formula 3-5 must be absolute temperatures.
Temperatures in ºF can be converted to absolute
temperatures in degrees Rankine (ºR) by adding 458
degrees. Thus, 32ºF becomes 32 + 458 = 490ºR.
The mathematical result of the calculation (T 1 4 -T 2 4 )
changes much more than the simple ∆T term used in the
formulas for conduction and convection. For example,
consider two surfaces exchanging radiant heat with
temperatures of 100ºF and 80ºF. These temperatures
would convert to 558ºR and 538ºR. The difference between
these temperatures would be only 20ºR, the same as the
difference between 100ºF and 80ºF. However, when these
Rankine temperatures are used in Formula 3-5 the resulting
number for the term (T 1 4 -T 2 4 ) is 131,700,000ºR 4 .
Due to mathematical complexities, as well as variability
or uncertainty in properties such as surface emissivities,
theoretical calculations of radiant heat transfer are often
limited to relatively simple situations.
This formula is more complex that those used for estimating
conduction and convection heat transfer.
⎛
v = 0.408 ⎞
⎝
⎜
⎠
⎟ f
d 2
The value of the “shape factor” (F 12 ) is a number between
0 and 1.0. It’s determined based on the relative angle and
distance between the two surfaces exchanging radiant
heat. Heat transfer textbooks give specific methods for
finding values of the shape factor (F 12 ) for different surfaces
⎛
and orientations. v = For 0.408 ⎞
⎛
two parallel planes having infinite
width and depth, the ⎝
⎜
value of ⎠
⎟ f = 0.408
⎞
⎜ ⎟
the shape factor (F 12 ) is 1.0.
d 2
ft
5 = 3.1
⎝
⎜ ( 0.811) 2
⎠
⎟ sec
Factors e 1 and e 2 are the emissivities of surfaces 1 and 2.
Emissivity is a surface property determined experimentally
based on how well the surface emits thermal radiation.
It must be a number between 0 and 1. A high value
indicates that the surface is a good emitter, and vice versa.
Emissivity values for various surfaces can be found in
references such as heat transfer handbooks. Interestingly,
the emissivity of a surface is not necessarily correlated with
its color. A rough metal surface coated with white enamel
paint has an emissivity of 0.91, and a flat black painted
surface has an emissivity of 0.97. Freshly fallen snow can
have an emissivity over 0.90. The emissivity of a polished
copper surface is 0.023, while a heavily oxidized copper
surface has an emissivity of 0.78. Most highly polished
metal surfaces have low emissivities, and thus would not
be good choices for the surface of a hydronic heat emitter
that’s expected to radiate heat into a room.
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