9-19E A hot plate with an insulated back is considered. The rate of heat loss by natural convection is to be determined for different orientations.

Assumptions 1 Steady operating conditions exist. 2 Air is an ideal gas with constant properties. 3 The local atmospheric pressure is 1 atm.

Properties The properties of air at 1 atm and the film temperature of (Ts+T)/2 = (130+75)/2 = 102.5F are (Table A-15)

Analysis(a) When the plate is vertical, the characteristic length is the height of the plate. Then,

and

(b) When the plate is horizontal with hot surface facing up, the characteristic length is determined from

.

Then,

and

(c) When the plate is horizontal with hot surface facing down, the characteristic length is again and the Rayleigh number is . Then,

and

9-49 Water in a tank is to be heated by a spherical heater. The heating time is to be determined.

Assumptions 1 Steady operating conditions exist. 2 The temperature of the outer surface of the sphere is constant.

Properties Using the average temperature for water (15+45)/2=30 as the fluid temperature, the properties of water at the film temperature of (Ts+T)/2 = (85+30)/2 = 57.5C are (Table A-9)

Also, the properties of water at 30C are (Table A-9)

AnalysisThe characteristic length in this case is Lc = D = 0.06 m. Then,

The rate of heat transfer by convection is

The mass of water in the container is

The amount of heat transfer to the water is

Then the time the heater should be on becomes

9-57 A heat sink with equally spaced rectangular fins is to be used to cool a hot surface. The optimum fin height and the rate of heat transfer from the heat sink are to be determined.

Assumptions 1 Steady operating conditions exist. 2 Air is an ideal gas with constant properties. 3 The atmospheric pressure at that location is 1 atm.

Properties The properties of air at 1 atm and 1 atm and the film temperature of (Ts+T)/2 = (65+25)/2 = 45C are (Table A-15E)

AnalysisThe characteristic length in this case is the height of the surface Lc = L = 0.18 m. Then,

The optimum fin spacing is

The heat transfer coefficient for this optimum fin spacing case is

The criteria for optimum fin height H in the literature is given by (not in the text) where Ac/p  t/2 for rectangular fins. Therefore,

The number of fins and the total heat transfer surface area is

Then the rate of natural convection heat transfer becomes

9-67 The absorber plate and the glass cover of a flat-plate solar collector are maintained at specified temperatures. The rate of heat loss from the absorber plate by natural convection is to be determined.

Assumptions 1 Steady operating conditions exist. 2 Air is an ideal gas with constant properties. 3 Heat loss by radiation is negligible. 4 The air pressure in the enclusure is 1 atm.

Properties The properties of air at 1 atm and the average temperature of (T1+T2)/2 = (80+40)/2 = 60C are (Table A-15)

AnalysisFor , we have horizontal rectangular enclosure. The characteristic length in this case is the distance between the two glasses Lc = L = 0.025 m Then,

Then

For , we obtain

For , we have vertical rectangular enclosure. The Nusselt number for this geometry and orientation can be determined from (Ra = 3.689104 - same as that for horizontal case)

DiscussionCaution is advised for the vertical case since the condition H/L < 40 is not satisfied.