Estimate the heat loss due to free convection from a pipe
surface that is at 110 degrees Fahrenheit to ambient air at 90 degrees
Fahrenheit. Outside diameter of the
pipe is 1.66 inch.
Solution:
Heat transfer due to free convection is given in the terms
of correlations that are dependent upon geometry and configuration of the
system under consideration. Heat
transfer coefficient is given through Nusselt equation that relates heat
transfer coefficient to physical properties of the liquid and gases, and the
geometry of the system. Nusselt number
is a combination of heat transfer coefficient (h) characteristic length
(L) and thermal conductivity (k) of gas. This dimensionless group is a function of
other dimensionless number, Grashof number, Gr, and Prandtl Number (Pr).
Grashof number is a function of characteristics length, L, density of
gas, r,
coefficient of thermal expansion (b), temperature difference (DT), and viscosity (m).
Prandtl Number is a function of specific heat, viscosity and
thermal conductivity. Please note that
the properties of the fluid have to be obtained at average film temperature.
There is another dimensionless group, used in these kinds of
calculations, and that is Raleigh Number or Y.
This is simply a product of Grahsof and Prandtl numbers.
Film temperature can be obtained by taking an average of the
surface and ambient temperatures. The
film temperature is 100 degrees Fahrenheit and the properties of air at this
temperature are listed as: viscosity 0.0460 pound per foot per hour, density
0.071 pounds per cubic foot, thermal conductivity 0.0156 BTU per hour per foot
per degree Fahrenheit, and specific heat 0.24 BTU per pound per degree
Fahrenheit.
Beta, coefficient of thermal expansion, for ideal gases is
given as 1 over T where T is the absolute temperature. It can be found to be equal to 0.00179 per
degree Rankine. Acceleration due to
gravity is 32.174 ft per second squared or 4.17 times 10 to the power 8 foot
per hour squared. Note that the DT is 110
minus 90 equals 20 degrees Fahrenheit.
With these values, Grashof number is calculated to be 9.5
times 10 to the power 4, whereas Prandtl number is 0.708. This gives a value of Raleigh number as
6.723 times 10 to the power 4.
Nusselt number is given as a times Y to the power b. For horizontal pipe, value of parameter a
equals 0.525 and b equals 0.25 (See Perry, Table 10.1). Nusselt number is calculated as 8.454. Nusselt number is h times L over k. In this case, L is replaced by d and the
heat transfer coefficient is determined to be 0.95 BTU per hour per foot
squared per degree Fahrenheit.
Heat transfer area is given by pi d times Lp,
where Lp is the pipe length.
A different notation has been used to differentiate between
characteristic length and the pipe length.
Heat transfer area is 0.436 square foot.
Heat transfer is given as h A times DT or 8.29
BTU per hour.