ME 353 Heat Transfer 1

M.M. Yovanovich

FEF96P2.MWS

_________________________________________

Problem 2 of Final Examination, December 9, 1996.

See the problem statement.

__________________________________________

> restart:

Define the system parameters.

> case1:=
(D = 0.050, Ts = 140 + 272, Tinfty = 25 + 273,
k = 0.0261, nu = 15.71*10^(-6), Pr = 0.71,
U = 0.5, g = 9.81, beta = 1/(25 + 273));

Define the forced and natural convection correlation

equations for the isothermal sphere.

> NuDFC:= 2 + 0.6*ReD^(1/2)*Pr^(1/3);

> ReD:= U*D/nu; #Reynolds number for the sphere.

> NuDNC:= 2 + FPr*RaD^(1/4);

> FPr:= 0.589/(1 + (0.469/Pr)^(9/16))^(4/9);
#Prandtl number function.

> RaD:= g*beta*(Ts - Tinfty)*D^3/(nu*alpha);
#Rayleigh number.

> alpha:= nu/Pr;
#Thermal diffusivity from definition of Prandtl number.

Compute the Reynolds and Rayleigh numbers for this problem.

> ReD1:= evalf(subs(case1, ReD), 6);

> RaD1:= evalf(subs(case1, RaD), 6);

The forced and natural convection flows are laminar.

Compute the Nusselt numbers for pure forced and pure natural

convection from the isothermal sphere.

> NuDFC1:= evalf(subs(ReD = ReD1, case1, NuDFC), 4);

> NuDNC1:= evalf(subs(RaD = RaD1, case1, NuDNC), 4);

Define the average heat transfer coefficient with respect to the Nusselt

number.

> h:= NuD*k/D;

Compute the forced and natural convection heat transfer coefficients.

> hFC:= evalf(subs(NuD = NuDFC1, case1, h), 4);

> hNC:= evalf(subs(NuD = NuDNC1, case1, h), 4);

Observe that the natural and forced convection heat transfer

coefficients are comparable.

Define the convection heat transfer rate from the sphere into the fluid,

i.e., Newton's Law of Cooling.

> h:= 'h':

> Q:= h*A*(Ts - Tinfty);

> A:= Pi*D^2;

Compute the pure forced and natural convection heat transfer rates from

the isothermal sphere.

> QFC:= evalf(subs(h = hFC, case1, Q), 4);

> QNC:= evalf(subs(h = hNC, case1, Q), 4);

Compute the maximum radiation heat transfer rate from the isothermal

sphere to its surroundings. Assume black-blody radiation.

Stefan-Boltzmann net radiation expression.

> Qrad:= sigma*(T1^4 - T2^4)*A;

> Qrad1:=
evalf(subs(sigma = 5.67*10^(-8), T1 = 140 + 273,
T2 = 25 + 273, case1, Qrad), 4);

Observe that for this problem the radiative heat transfer rate

is comparable to the natural and forced convective heat transfer

rates.

Compute the total heat transfer rate from the sphere by forced

and natural convection plus radiation.

> Q_FC_NC_Rad:= evalf(QFC + QNC + Qrad1, 4);

Compute the conduction through the support, etc.

> Qcond:= 50 - %;

Observation:

The forced and natural convection correlation equations contain

the zero flow (diffusive limit), i.e., NuD = 2.

It should not be included twice in the calculations.

Therefore the total NuD due to forced + natural + radiation

should be decreased by 2 and the NuD for conduction should

be increased by 2.

The conduction heat transfer rate is greater than the value given above.