Growers use giant fans to prevent grapes from freezing when the effective sky temperature is low. The grape, which may be viewed as a thin skin of negligible thermal resistance enclosing a volume of sugar water, is exposed to ambient air and is irradiated from the sky above and ground below. Assume the grape to be an isothermal sphere of 15-mm diameter, and assume uniform blackbody irradiation over itstop and bottom hemispheres due to emission from the sky and the earth, respectively. (a) Derive an expression for the rate of change of the grape temperature. Express your result in terms of a convection coefficient and appropriate temperatures and radiative quantities. (b) Under conditions for which T s k y = 235 K , T ∞ = 273 K , and the fan is off ( V = 0 ) , determine whether the grapes will freeze. To a good approximation, the skin emissivity is 1 and the grape thermophysical properties are those of sugarless water. However, because of the sugar content, the grape freezes at − 5 ∘ C . (c) With all conditions remaining the same, except that the fans are now operating with V = 1 m / s , will the grapes freeze?
Growers use giant fans to prevent grapes from freezing when the effective sky temperature is low. The grape, which may be viewed as a thin skin of negligible thermal resistance enclosing a volume of sugar water, is exposed to ambient air and is irradiated from the sky above and ground below. Assume the grape to be an isothermal sphere of 15-mm diameter, and assume uniform blackbody irradiation over itstop and bottom hemispheres due to emission from the sky and the earth, respectively. (a) Derive an expression for the rate of change of the grape temperature. Express your result in terms of a convection coefficient and appropriate temperatures and radiative quantities. (b) Under conditions for which T s k y = 235 K , T ∞ = 273 K , and the fan is off ( V = 0 ) , determine whether the grapes will freeze. To a good approximation, the skin emissivity is 1 and the grape thermophysical properties are those of sugarless water. However, because of the sugar content, the grape freezes at − 5 ∘ C . (c) With all conditions remaining the same, except that the fans are now operating with V = 1 m / s , will the grapes freeze?
Solution Summary: The author explains the expression for the rate of change grape temperature.
Growers use giant fans to prevent grapes from freezing when the effective sky temperature is low. The grape, which may be viewed as a thin skin of negligible thermal resistance enclosing a volume of sugar water, is exposed to ambient air and is irradiated from the sky above and ground below. Assume the grape to be an isothermal sphere of 15-mm diameter, and assume uniform blackbody irradiation over itstop and bottom hemispheres due to emission from the sky and the earth, respectively.
(a) Derive an expression for the rate of change of the grape temperature. Express your result in terms of a convection coefficient and appropriate temperatures and radiative quantities.
(b) Under conditions for which
T
s
k
y
=
235
K
,
T
∞
=
273
K
, and the fan is off
(
V
=
0
)
, determine whether the grapes will freeze. To a good approximation, the skin emissivity is 1 and the grape thermophysical properties are those of sugarless water. However, because of the sugar content, the grape freezes at
−
5
∘
C
.
(c) With all conditions remaining the same, except that the fans are now operating with
V
=
1
m
/
s
, will the grapes freeze?
1. A small gray sphere, with an emissivity coefficient of 0.5 and
a surface temperature of 537°C, is located in a black body
wrap with a temperature of 35°C. For this system, calculate
the net rate of heat transfer per unit of surface area of the
sphere.
2. Gaseous oxygen is maintained at pressures of 2 atm and 1
atm on the opposite sides of a rubber membrane, which has
a thickness of 0.5 mm, and the entire system is at
25°C. What is the diffusive mass flow of gas through the
membrane?
DAB=0.21x10^-9 m^2/s; O = 16 g/mol
3. Pure oxygen gas at 2 bar and 25°C is flowing through a
rubber hose of 10 m long, with 40 mm internal diameter and
2 mm wall thickness. The external surface is exposed to an
air stream in which the partial pressure of the gas is 0.1 bar.
The diffusivity and solubility of the gas in the hose material
are 0.21x10^-9 m^2/s and 3.12x10^-3 kmol/m^3.bar.
respectively. Determine the mass rate at which the gas leaks
out of the hose.
4. Consider the diffusion of gaseous…
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