Natural Convection: Two Concentric Isothermal Spheres

M.M. Yovanovich

NC2SPH2.MWS

Natural convection across gap formed by two concentric, isothermal spheres.

Simpler form of the Raithby-Hollands set of correlation equations is

recommended.

> restart:

System parameters.

> syspar:= (g=9.81, Di=40/1000, Do=80/1000, Ti=360, To=300);

Dry air properties correlation equations for temperatures from 250 to 450 K.

> 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:

Heat transfer relations.

> nc2sph:= Q - h*Ai*(Ti-To)=0:

> h:= k(T)*NuDi/Di:

> Ai:= Pi*Di^2:

> NuDi:= 0.74*(Pr(T)/(0.861 + Pr(T))^(1/4)*RaDi^(1/4)/(1+(Di/Do)^(7/5))^(5/4)):

> RaDi:= g*beta*(Ti-To)*Di^3/(alpha(T)*nu(T)):

> beta:= 1/Tf: Tf:= (Ti+To)/2:

Diffusive limit.

> SstarDi:= 2/(1- Di/Do):

> SstarDi1:= evalf(subs(syspar, SstarDi),5);

> syspar;

> Tf1:= evalf(subs(T=Tf, syspar, Tf),4);

> airprops:= evalf(subs(T=Tf1, syspar, [k=k(T), rho=rho(T), cp=cp(T), mu=mu(T), nu=nu(T), alpha=alpha(T), Pr=Pr(T)]),4);

> NuDi1:= evalf(subs(T=Tf, syspar, NuDi),5);

Since the boundary layer value is larger than the diffusive limit, according to the Raithby-Hollands model, one must use the larger value.

Solve for Q given the cylinder temperatures.

> Q1:= evalf(solve(subs(T=Tf, syspar, nc2sph), Q), 5);

Solve for given Q.

> syspar:= (g=9.81, Di=40/1000, Do=80/1000, To=300);

> Q:= 1.75:

> Ti1:= evalf(fsolve(subs(T=Tf, syspar, nc2sph), Ti, 250..450),5);

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