Natural Convection from Isothermal Cube

M.M. Yovanovich

NCCUBE3.MWS

Natural convection from a horizontal, isothermal cube. Use the correlation based on the square root of the total surface area. The dimensionless shape factor is 3.388 and the body-gravity function value is 0.985.

> restart:

Air properties correlation equations.

>

> k:= T->-.2571428571e-7*T^2+.9300000000e-4*T+.6685714286e-3:

> rho:= T->.8997142857e-5*T^2-.9361400000e-2*T+3.168594286:

> cp:= T->3/7000*T^2-113/500*T+7251/7:

> mu:=T->-.3000000000e-10*T^2+.6654000000e-7*T+.1200000000e-5:

> nu:=T->.1005714286e-9*T^2+.3444000000e-7*T-.3466857143e-5:

> alpha:=T->.1600000000e-9*T^2+.4480000000e-7*T-.5320000000e-5:

> Pr:=T-> -.1700000000e-3*T+.7601000000:

> airprops:= [k=k(T), rho=rho(T), cp=cp(T), mu=mu(T), nu=nu(T), alpha=alpha(T), Pr=Pr(T)]:

Newton's Law of Cooling.

> nccube:= Q-h*A*(Tw-Tinfty)=0;

[Maple Math]

> h:= k(T)*NusqrtA/sqrt(A):

> A:= 6*S^2;

[Maple Math]

Correlation equation for a horizontal cube.

> NusqrtA:= SstarsqrtA + 0.670/(1+(0.5/Pr)^(9/16))^(4/9)*GsqrtA*RasqrtA^(1/4):

> Pr:= Pr(T):

> RasqrtA:= g*beta*(Tw-Tinfty)*(sqrt(A))^3/(alpha(T)*nu(T)):

Specify the system parameters.

> g:= 9.81: beta:= 1/Tinfty: SstarsqrtA:= 3.388: GsqrtA:= 0.985:

> Tinfty:= 300: T:= (Tw+Tinfty)/2: S:= 120/1000:

Specify the heat transfer rate [Maple Math] and solve for the surface temperature [Maple Math] .

> Q:= 10.5: Tw:= 'Tw':

> Tw:= evalf(fsolve(nccube, Tw, 0..1000), 5);

[Maple Math]

Check Rayleigh number.

> evalf(RasqrtA, 4);

[Maple Math]

The fluid flow is laminar.

Specify the wall temperature [Maple Math] and solve for the heat transfer rate [Maple Math] .

> Q:='Q': Tw:='Tw': Tw:= 365:

> Q:= evalf(fsolve(nccube, Q, 0..100), 5);

[Maple Math]

> evalf(RasqrtA, 5);

[Maple Math]

The flow is laminar.