Question 10.35: KNOWN: Copper sphere, 10 mm diameter, initially at a prescri......
KNOWN: Copper sphere, 10 mm diameter, initially at a prescribed elevated temperature is quenched in a saturated (1 atm) water bath.
FIND: The time for the sphere to cool (a) from \mathrm T_{\mathrm i} = 130 to 110°C and (b) from \mathrm T_{\mathrm i} = 550°C to 220°C.
ASSUMPTIONS: (1) Sphere approximates lumped capacitance, (2) Water saturated at 1 atm.
PROPERTIES: Table A-1, Copper: ρ = 8933 kg/m³; Table A.11, Copper (polished) : ε = 0.04, typical value; Table A.4, Water: as required for the pool boiling correlations.
SCHEMATIC:
ANALYSIS: Treating the sphere as a lumped capacitance and performing an energy balance, see Eq. 5.14,
\dot{ E }_{\text {in }}-\dot{ E }_{\text {out }}=\dot{ E }_{ st } ~~~\quad~~~ -q _{ s }^{\prime \prime} \cdot A _{ s }=\rho c \forall \frac{ dT }{ dt } (1,2)
For the sphere, \mathrm V = \pi\mathrm D^3/6~\text{and}~\mathrm A_{\mathrm s} = \pi\mathrm D^2. Using the IHT Lumped Capacitance Model to solve this differential equation, we need to specify (1) the specific heat of the copper sphere as a function sphere temperature; use IHT Properties Tool, Copper; and (2) the heat flux, \mathrm q_{\mathrm s}^{\prime\prime}, associated with the pool boiling processes; use IHT Correlations Tool, Boiling:
(a) Cooling from \mathrm T_{\mathrm i} =130° to 110°: Nucleate pool boiling, Rohsenhow correlation, Eq. 10.5,
(b) Cooling from \mathrm T_{\mathrm i} = 550 to 220°C : Film Pool Boiling, Eq. 10.9 with C = 0.67 (sphere).
The thermophysical properties for water required of the correlations are provided by the IHT Tool, Properties-Water. The specific heat of copper as a function of sphere temperature is provided by the IHT Tool, Properties-Copper. The temperature-time histories for each of the cooling processes are plotted below.
Using the Explore feature in the IHT Plot Window, the elapsed times for the quench process were found as:
COMMENTS: (1) Comparing the elapsed times for the two processes, the nucleate pool boiling process cools 20°C in 0.76s (26.3°C/s) vs. 330°C in 13.5s (24.4°C/s) for the film pool boiling process.
(2) The IHT Workspace used to generate the temperature-time history for the nucleate pool boiling process is shown below.
| // Correlations Tool – Boiling, Nucleate Pool Boiling, Heat flux qs” = qs_dprime_NPB(Csf,n,rhol,rhov,hfg,cpl,mul,Prl,sigma,deltaTe,g) // Eq 10.5 g = 9.8 // Gravitational constant, m/s^2 deltaTe = Ts – Tsat // Excess temperature, K Ts = Ts_C + 273 // Surface temperature, K //Ts_C = 130 Tsat = 100 + 273 // Saturation temperature, K /* Evaluate liquid(l) and vapor(v) properties at Tsat. From Table 10.1 (Fill in as required), */ // fluid-surface combination: Csf = 0.013 // Polished copper-water combination, Table 10.1 n = 1.0 /* Correlation description: Heat flux for nucleate pool boiling (NPB), water-surface combination (Cf,n), Eq 10.5, Table 10.1 . See boiling curve, Fig 10.4 . */ // Properties Tool- Water: // Water property functions :T dependence, From Table A.6 // Units: T(K), p(bars); xv = 1 // Quality (0=sat liquid or 1=sat vapor) rhov = rho_Tx(“Water”,Tsat,xv) // Density, kg/m^3 hfg = hfg_T(“Water”,Tsat) // Heat of vaporization, J/kg sigma = sigma_T(“Water”,Tsat) // Surface tension, N/m (liquid-vapor) // Water property functions :T dependence, From Table A.6 // Units: T(K), p(bars); xl = 0 // Quality (0=sat liquid or 1=sat vapor) rhol = rho_Tx(“Water”,Tsat,xl) // Density, kg/m^3 cpl = cp_Tx(“Water”,Tsat,xl) // Specific heat, J/kg·K mul = mu_Tx(“Water”,Tsat,xl) // Viscosity, N·s/m^2 Prl = Pr_Tx(“Water”,Tsat,xl) // Prandtl number // Lumped Capacitance Model: /* Conservation of energy requirement on the control volume, CV. */ Edotin – Edotout = Edotst Edotin = 0 Edotout = As * ( + qs” ) Edotst = rho * vol * cp * Der(Ts,t) /* The independent variables for this system and their assigned numerical values are */ As = pi * D^2 / 4 // surface area, m^2 vol = pi * D^3 / 6 // volume, m^3 D = 0.01 rho = 8933 // density, kg/m^3 // Properties Tool – Copper // Copper (pure) property functions : From Table A.1 // Units: T(K) cp = cp_T(“Copper”,Ts) // Specific heat, J/kg·K |
| Quench process | \mathrm T_{\mathrm i} – \mathrm T_{\mathrm f} (°C) | Δt(s) |
| Nucleate pool boiling | 130-110 | 0.76 |
| Film pool boiling | 550-220 | 13.5 |
