CHE312 PROBLEM SET 1 1 (1371) THE ENERGY CONSUMPTION

Problem set #1

CHE312: Problem set #1

1. (1.37^¹) The energy consumption associated with a home water heater has two components: (i) the energy that must be supplied to bring the temperature of ground water to the heater storage temperature, as it is introduced to replace hot water that has been used, and (ii) the energy needed to compensate for heat losses incurred while the water is stored at the prescribed temperature. In this problem, we will evaluate the first of these components for a family of four, whose daily hot water consumption is approximately 100 gallons. If ground water is available at 15^oC, what is the annual energy consumption associated with heating the water to a storage temperature of 55^oC? For a unit electrical power cost of $0.08/kWh, what is the annual cost associated with supplying hot water by means of (a) electric resistance heating or (b) a heat pump having a COP of 3 and a compressor efficiency of 85% (conversion from electrical energy to mechanical work).

Water density = 993 kg/m³ at 0.5(15 + 55) = 35^oC = 308 K, C_pw = 4.187 kJ/kgK, 100 gal = 0.3785 m³

(a) Annual cost associated with supplying hot water by means of electric resistance heating

Resistance heating cost = (6369 kWh)($0.08/kWh) = $510

(b) Annual cost associated with supplying hot water by means of a heat pump

Heat pump cost = (2498 kWh)($0.08/kWh) = $200

Operating costs are lower for a heat pump, capital costs are higher.

2. (1.49¹) Liquid oxygen, which has a boiling point of 90^oK and a latent heat of vaporization of 214 kJ/kg, is stored in a spherical container whose outer surface is of 500-mm diameter and at a temperature of 10^oC. The container is housed in a laboratory whose air and walls are at 25^oC.

(a) If the surface emissivity is 0.20 and the heat transfer coefficient associated with free convection at the outer surface of the container is 10 W/m²K, what is the rate, in kg/s, at which oxygen vapor must be vented from the system?

(b) Moisture in the ambient air will result in frost formation on the container, causing the surface emissivity to increase. Assuming the surface temperature and convection coefficient to remain at 10^oC and 10 W/m²K, respectively, compute the oxygen evaporation rate (kg/s) as a function of surface emissivity over the range 0.2  0.94.

(a) = 1.4110^-3 kg/s

(b) Compute the oxygen evaporation rate (kg/s) as a function of surface emissivity over the range 0.2  0.94.

CHE312 PROBLEM SET 1 1 (1371) THE ENERGY CONSUMPTION

3. (1.55¹) The roof of a car in a parking lot absorbs a solar radiant flux of 800 W/m², while the underside is perfectly insulated. The convection coefficient between the roof and the ambient air is 12 W/m²K.

(a) Neglecting radiation exchange with the surroundings, calculate the temperature of the roof under steady-state conditions if the ambient air temperature is 20^oC.

(b) For the same ambient air temperature, calculate the temperature of the roof if its surface emissivity is 0.8.

(c) The convection coefficient depends on airflow conditions over the roof, increasing with increasing air speed. Compute and plot the roof temperature as a function of h for 2 h 200 W/m²K.

Solution

(a) 86.7^oC

(b) 47^oC

CHE312 PROBLEM SET 1 1 (1371) THE ENERGY CONSUMPTION

4. (1.59¹) A surface whose temperature is maintained at 400^oC is separated from an air flow by a layer of insulation 25 mm thick for which the thermal conductivity is 0.1 W/mK. If the air temperature is 35^oC and the convection coefficient between the air and the outer surface of the insulation is 500 W/m²K, what is the temperature of this outer surface?

T₂ = 37.9^oC

5. (1.61¹) An experiment to determine the convection coefficient associated with airflow over the surface of a thick steel casting involves insertion of thermocouples in the casting at distances of 10 and 20 mm from the surface along a hypothetical line normal to the surface. The steel has a thermal conductivity of 15 W/mK. If the thermocouples measure temperatures of 50 and 40^oC in the steel when the air temperature is 100^oC, what is the convection coefficient?

h = 15,000/(100  60) = 375 W/m²K

6. (1.65¹) Plate glass at 600^oC is cooled by passing air over its surface such that the convection heat transfer coefficient is h = 5 W/m²K. To prevent cracking, it is known that the temperature gradient must not exceed 15^oC/mm at any point in the glass during the cooling process. If the thermal conductivity of the glass is 1.4 W/mK and its surface emissivity is 0.8, what is the lowest temperature of the air that can initially be used for the cooling? Assume that the temperature of the air equals that of the surroundings.

T_ = 618 K = 345^oC

7.^² A cylinder of radius r_o, length L, and thermal conductivity k is immersed in a fluid of convection coefficient h and unknown temperature T_. At a certain instant the temperature distribution in the cylinder is T(r) = a + br², where a and b are constants. Obtain expressions for the heat transfer rate at r_o and the fluid temperature.