0 m of water above the insulating layer has a transmittance

0 m of water above (the insulating layer) has a transmittance r 0.7, and its density increases downwards so that convection does not occur (see Fig. 4.6). The designer hopes to maintain the storage layer at 80 C. The temperature at the surface of the pond is 270C (da and night) (a) Calculate the thermal resistance of the insulating layer, and compare it with the top resistance of a typical flat-plate collector. (b) Calculate the thermal resistance of a similar layer of fresh water subject to free convection. Compare this value with that in (a), and comment on any improvement. (c) The density of NaCl solution increases by 0.75 g/liter for every 1.0 g of NaCl added to 1.0 kg H2O. A saturated solution of NaCl con tains about 370 g NaCl per kg H2O. The volumetric coefficient of thermal expansion of NaCl solution is about 4x10-4/K. Calculate the minimum concentration C of NaCl required in the storage layer to suppress convection, assuming the water layer at the top of the pond contains no salt. How easy is it to achieve this concentration in practice? (d) Calculate the characteristic time scale for heat loss from the storage ayer, through the resistance of the insulating layer. If the tempera ture of the storage layer is 800C at sunset (6 p.m.) what is its tem perature at sunrise (6 a.m.)? (e) Diffusion of a solute depends on its molecular diffusivity D analogous to thermal diffusivity of SR3-3. For NaCl in water 1.5 x 10 m\'s 1.The pond is set up with the storage layer having twice the critical concentration of NaC i e. double the value C cal culated in (c). Estimate the time for molecular diffusion to decrease this concentration to Cman Note: Molecular diffusion of salt (solute) from a region of strong concentration to a weak concentration is analogous to the diffusion of heat from a region of high temperature to a region of low tempera ture by conduction, as described in SR3.3. In fact, the same equa tions apply with molecular diffusivity Din place of thermal diffusivity K and concentration Cin place of temperature T. (f) According to your answers to (c)-(e), discuss the practicability of building such a pond, and the possible uses to which it could be put.

Solution

An idealized solar pond measures 100m×100m×1.2m. The bottom 20 cm (the storage layer) has an effective absorptance = 0.7. The 1.0m of water above (the insulating layer) has a transmittance = 0.7, and its density increases downwards so that convection does not occur. The designer hopes to maintain the storage layer at 80^oC. The temperature at the surface of the pond is 27^oC (day and night).

a Calculate the thermal resistance of the insulating layer, and compare it with the top resistance of a typical flat plate collector.

Answer A:

r_n= 1.7m²KW¹ > 0.4m²KW¹ (for good flat plate)

b Calculate the thermal resistance of a similar layer of fresh water, subject to free convection. Compare this value with that in (a), and comment on any improvement.

Answer B:

r_v= 0.0018m²KW¹ <<r_n of (a).

c The density of NaCl solution increases by 0.75 g per litre for every 1.0 g of NaCl added to 1.0 kg H2O. A saturated solution of NaCl contains about 370 g of NaCl per kg H2O. The volumetric coefficient of thermal expansion of NaCl solution is about 4×10⁴ K¹. Calculate the minimum concentration Cmin of NaCl required in the storage layer to suppress convection, assuming the water layer at the top of the pond contains no salt. How easy is it to achieve this?

Concentration in practice?

Answer C:

c 30 g NaCl per kg H2O<< saturated concentration.

d Calculate the characteristic time scale for heat loss from the storage layer, through the resistance of the insulating layer. If the temperature of the storage layer is 80^oC at sunset (6 p.m.) what is its temperature at sunrise (6 a.m.)?

Answer D:

RC 1.3×10⁶ s = 15 day (neglected bottom loss)

Sunrise temp =T_a+(T_storeT_a)₀ exp(t/(RC))= 6+(80-6)EXP(-80/1.3×10⁶)=78.3^oC.

e The molecular diffusivity of NaCl in water is 1.5×10⁵ cm² s¹. The pond is set up with the storage layer having twice the critical concentration of NaCl, i.e. double the value Cmin calculated in (c). Estimate the time for molecular diffusion to lower this concentration to Cmin.

Answer E:

Diffusion time X²/D 5y.

f According to your answers to (c)–(e), discuss the practicability of building such a pond, and the possible uses to which it could be put.

Answer F:

Salt gradient easily established and maintained against diffusion. Heat well retained, therefore theoretically viable.