Estimate the heat transfer when water is flowing at a rate
of 10,000 pounds per hour through a pipe at 80 degrees Fahrenheit and is heated
to 120 degrees Fahrenheit. Inside
diameter of the pipe is 1.334 inch.
Viscosity of water at wall temperature is 1.406 pounds per foot per
hour.
Solution:
Heat transfer due to forced convection is also given in the
terms of correlations. Heat transfer
coefficient is given by Nusselt equation that relates heat transfer coefficient
to physical properties of the liquid and gases. Nusselt number is a combination of heat transfer coefficient (h)
characteristic length (L) and thermal conductivity (k) of the
fluid. This dimensionless group is a
function of other dimensionless number, Graetz number (Gz), which in turn
depends upon Reynolds Number (Re), Prandtl Number (Pr), and
dimensionless characteristic length. It
also depends upon whether fluid is flowing inside the pipe or in the
annulus. Reynolds number is a function
of velocity (V), density, r, and viscosity
(m) of the
fluid. 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 bulk temperature.
Bulk temperature can be obtained by taking an average of the
inlet and outlet temperatures. The film
temperature is 100 degrees Fahrenheit and the properties of water at this
temperature are listed as: viscosity 1.406 pound per foot per hour, density
62.0 pounds per cubic-foot, thermal conductivity 0.363 BTU per hour per foot
per degree Fahrenheit, and specific heat 0.998 BTU per pound per degree
Fahrenheit.
Given the mass flow rate, we can find velocity through the pipe, once cross sectional area of the pipe is known. For a circular pipe, this value is given by pi over 4 d squared and equals 0.00971 square-foot. Dividing mass flow rate by density and cross sectional area, we get velocity of 1.6618 times 10 to the power 4 feet per hour.
Reynolds number is V time d time rho over mu. Plugging in their values, we get Reynolds number as 6.9415 times 10 to the power 4. Similarly, we can find another dimensionless group, Prandtl number, which is given by c time mu over k. Its value is found to be 4.54.
It can be seen that Reynolds number is greater than 10,000. We can choose appropriate correlation. In this case, Nusselt number is given by
0.023 times Reynolds number to the power (0.8) time Prandtl number to the power
(0.333) time phi. In this expression
phi is a viscosity correction-factor.
Uncorrected Nusselt number is calculated as 284.2. Nusselt number is h time L over k. In this case, L is replaced by d and the
uncorrected heat transfer coefficient is determined to be 927.9 BTU per hour
per foot squared per degree Fahrenheit.
Viscosity correction factor is (mu over muw) to
the power 0.14. In this expression, mu
is the absolute viscosity at bulk temperature, and subscript w denotes
wall. So, muw is the
viscosity at wall temperature. This
correction actor is found to be 1.0227.
Corrected heat transfer coefficient is 927.9 times 1.0227
equals 948.9 BTU per hour per foot squared per degree Fahrenheit.