OFFSET
1,1
COMMENTS
Study of the heat transfer in cylindrical pipes with fully-developed laminar flow lwith constant inlet temperature and constant wall temperature (the Graetz-Nusselt problem) leads to the dimensionless equation 2 * (1-r^2) * dT/dz = 1/r * dT/dr + d^2T/dr^2 subject to the boundary conditions T(z=0) = 1, T(r=1) = 0, (dT/dr)(r=0) = 0.
The solution to this equation, obtained using separation of variables, is (where M is Kummer's M function and beta an eigenvalue) T = M(1/2 - 1/2 * beta, 1, 2*beta*r^2) * exp(- beta*r^2) * exp( - beta^2*z).
The first eigenvalue is calculated from the condition that the function value is zero for r=1: M(1/2 - 1/2 * beta[1], 1, 2*beta[1]) = 0.
The Nusselt number then is Nu = 2*beta[1]^2.
The Nusselt number was named after the German engineer Wilhelm Nusselt (1882-1957). - Amiram Eldar, May 18 2021
REFERENCES
Theodore L. Bergman and Adrienne S. Lavine, Fundamentals of Heat and Mass Transfer, Wiley, 2017, section 8.4, p. 491.
LINKS
Hans Dieter Baehr and Karl Stephan, Wärme- und Stoffübertragung, Springer Vieweg, 2013.
Wikipedia, Nusselt number.
EXAMPLE
Nu = 3.6567934577632923619...
MAPLE
fsolve(KummerM(1/2-1/2*beta, 1, 2*beta), beta=1..2)^2*2
MATHEMATICA
RealDigits[2 * (x /. FindRoot[Hypergeometric1F1[1/2 - x/2, 1, 2*x], {x, 1}, WorkingPrecision -> 120])^2, 10, 100][[1]] (* Amiram Eldar, May 18 2021 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Thomas König, Feb 19 2017
EXTENSIONS
a(100) corrected by Amiram Eldar, May 18 2021
STATUS
approved