Page 252 - Introduction to Computational Fluid Dynamics

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EXERCISES
Table 7.2: Properties for Exercise 1. May 25, 2005 11:14 231
ρ C ps C pl k s k l λ T w T m T l
2,180 1,549 1,549 0.49 0.49 1.37 ×10 5 200 220 230
2,800 900 1,100 200 90 3.9 ×10 5 573 933 933

6. In an energy storage device, a PCM is sandwiched between two streams of
◦
heat transfer fluid (HTF) as shown in Figure 7.10. The HTF flows at 200 C
2
with heat transfer coefficient 300 W/m -K. The PCM is initially in a sat-
◦
urated state (T m = 220 C) and its thickness is 8 cm. Estimate the time for
heat (sensible + latent) recovery and the quantity recovered. The PCM proper-
3
ties are as follows: ρ = 2,180 kg/m , C p = 1,549 J/kg-K, K = 0.49 W/m-K,
5
and λ = 1.37 × 10 J/kg.
7. Consider solidification of a PCM contained in a spherical vessel of radius R.
Initially, the PCM is at temperature T in = T m . The vessel wall temperature is
T w < T m and held constant with respect to time. Assuming only radial heat
transfer, the applicable energy equation is
∂(ρ h) ∂ ∂T
A = KA ,
∂t ∂r ∂r
2
where A = 4π r .
(a) Nondimensionalise this equation assuming constant properties.
(b) Discretise the equation and write a computer program to solve the discre-
tised equations. Use of a nonuniform grid with closer spacings near r = R
and r = 0 is desirable. Take ρ = C p = k = λ = 1, R = 1, and T m = 0 and
compute for T w =−0.1, −1.0, and, 10.0.
(c) Plot the variation of interface location R i /R as a function of dimensionless
time in each case and estimate total solidification time. Compare your
results with those of [7].

8. Repeat Exercise 7 for a superheated PCM so that T in > T m . Take T w =−1.0
and use three values of T in :0.1, 1, and 2.

h HTF 200°C

PCM 8 cm

h HTF 200°C

Figure 7.10. Phase-Change Energy Storage Device – Exercise 6.