ME 353 Heat Transfer 1

M.M. Yovanovich

EX3P5.MWS

Example 3.5 of 4th edition of Incropera and DeWitt.

Steady conduction through an insulated spherical shell with convection cooling at the outer boundary into air.

> restart:

Assumptions.

1. Steady-state conduction.

2. System is source free.

3. Constant properties.

4. One-dimensional conduction.

5. Negligible contact resistance between metallic shell and insulation.

6. Negligible radiative heat transfer between insulated surface and its environment.

System parameters.

> syspar:= (r1 = 0.25, r2 = 0.275, k = 0.0017, h2 = 20, Tf2 = 300, hfg = 2e5, rho = 804, Tf1 = 77);

Heat transfer rate through system.

> Q[sys]:= (Tf1 - Tf2)/Rtotal;

> Rtotal:= Rins + Rf2;

> Rins:= 1/(4*Pi*k)*(1/r1 - 1/r2);

> Rf2:= 1/(h2*A2); A2:= 4*Pi*r2^2;

> Qsys:= evalf(subs(syspar, Q[sys]), 6);

The heat transfer rate is from the surroundings through the two resistances which are connected in series into the liquid nitrogen.

Relative magnitude of the two resistances.

> Rins_val:= evalf(subs(syspar, Rins), 5);

> Rf2_val:= evalf(subs(syspar, Rf2), 5);

> evalf(Rins_val/Rf2_val, 4);

The thermal resistance of the insulation is approximately times greater than the film resistance.

Rate of nitrogen boil-off.

> Q_boil_off = hfg*mdot;

> mdot:= subs(Q_boil_off = Qsys, syspar, solve(%, mdot))*kg/s;