材料热力学习题答案2

The problems of the second law

1 The solar energy flux is about 4J cm2/min. in no focusing collector the surface temperature can reach a value of about 900℃. If we operate a heat engine using the collector as the heat source and a low temperature reservoir at 25℃, calculate the area of collector needed if the heat engine is to produce 1 horse power. Assume the engine operates at maximum efficiency. (2.1)

TH?TL(900?25)104W?QH??4?StTH900?27360SolutionP?W?746(W)tS?0.664(m2)

2 A refrigerator is operated by 0.25 hp motor. If the interior of the box is to be maintained at -20℃ ganister a maximum exterior temperature of 35℃, what the maximum heat leak (in watts) into the box that can be tolerated if the motor runs continuously? Assume the coefficient of performance is 75% of the value for a reversible engine. (2.2)

W?75%Solution:P?0.75TH?TLQHTH

TH?TLPLTLPL?4273?20??0.25?746?1144(W)335?20

3 suppose an electrical motor supplies the work to operate a Carnot refrigerator. The interior of the refrigerator is at 0℃. Liquid water is taken in at 0℃ and converted to ice at 0℃. To convert 1 g of ice to 1 g liquid. △H=334J/g is required. If the temperature outside the box is 20℃, what mass of ice can be produced in one minute by a 0.25 hp motor running continuously? Assume that the refrigerator is perfectly insulated and that the efficiencies involved have their largest possible value. (2.3)

P?TH?TLPLTLSolution: PL?273?0.25?746?334m 20M?60m?457(g)

4 under 1 atm pressure, helium boils at 4.126K. The heat of vaporization is 84 J/mol what size

motor (in hp) is needed to run a refrigerator that must condense 2 mol of gaseous helium at 4.126k to liquid at the same temperature in one minute? Assume that the ambient temperature is 300K and that the coefficient of performance of the refrigerator is 50% of the maximum possible. (2.4)

W?50%TH?TLQLTLTH?TL300?4.21684?2 PL??TL4.21660Solution: 0.5P'?P?P'?393(W)?0.52(hp)5 if a fossil fuel power plant operating between 540 and 50℃ provides the electrical power to

run a heat pump that works between 25 and 5℃, what is the amount of heat pumped into the house per unit amount of heat extracted from the power plant boiler.

(a) assume that the efficiencies are equal to the theoretical maximum values

(b) assume the power plant efficiency is 70% of maximum and that coefficient of performance of

the heat pump is 10% of maximum

(c) if a furnace can use 80% of the energy in fossil foe to heat the house would it be more

economical in terms of overall fissile fuel consumption to use a heat pump or a furnace ? do the calculations for cases a and b (2.5)

solution:

TH,1?TL,1(a)PPH,11?TH,1P2?TH,2?TL,2PH,2TH,2(b) PH,2?0.6286PH,1

P1?P2540?5025?5PH,1?PH,2540?273273?25PH,2?8.98PH,1(c)aisok,bisnot.6 calculate △U and △S when 0.5 mole of liquid water at 273 K is mixed with 0.5 mol of liquid water at 373 K and the system is allowed to reach equilibrium in an adiabatic enclosure. Assume that Cp is 77J /(mol K) from 273K to 373K (2.6) Solution:

?U?0(J) TT323323?S?n1CPln(E)?n2CPln(E)?0.5CPln()?0.5CPln()?0.933(J/K)T1T23732737 A modern coal burning power plant operates with a steam out let from the boiler at 540℃ and a

condensate temperature of 30℃.

(a) what is the maximum electrical work that can be produced by the plant per joule of heat

provided to the boiler?

(b) How many metric tons (1000kg) of coal per hour is required if the plant out put is to be

500MW (megawatts). Assume the maximum efficiency for the plant. The heat of combustion

of coal is 29.0 MJ/k g

(c) Electricity is used to heat a home at 25℃ when the out door temperature is 10℃ by passing

a current through resistors. What is the maximum amount of heat that can be added to the home per kilowatt-hour of electrical energy supplied? (2.7)

(a)W?Solution:(b)TH?TL540?30QH?1?0.89(J)TH540?30TL?500?3600TH?TL(c) W?TH?TLQHTH273?251?19.9(J)25?10

29.0mQH?m?69371(kg)?69.3(ton)

8 an electrical resistor is immersed in water at the boiling temperature of water (100℃) the electrical energy input into the resistor is at the rate of one kilowatt

(a) calculate the rate of evaporation of the water in grams per second if the water container is

insulated that is no heat is allowed to flow to or from the water except for that provided by the resistor

(b) at what rate could water could be evaporated if electrical energy were supplied at the rate of 1

kw to a heat pump operating between 25 and 100℃

data for water enthalpy of evaporation is 40000 J/mol at 100℃; molecular weight is 18g/mol; density is 1g/cm3 (2.8)

m40000?1000,m?0.45(g)18solution:

m100?273(b)40000?1000,m?2.23(g)18100?25(a)9 some aluminum parts are being quenched (cooled rapidly ) from 480℃ to -20℃ by immersing

them in a brine , which is maintained at -20℃ by a refrigerator. The aluminum is being fed into the brine at a rate of one kilogram per minute. The refrigerator operates in an environment at 30℃; that is the refrigerator may reject heat at 30℃. what is them minus power rating in kilowatts, of motor required to operate the refrigerator?

Data for aluminum heat capacity is 28J/mol K; Molecular weight 27g/mol (2.9)

100028?(480??20)27Solution:

TH?TL30??20PW?PL?PL?102474(W)?102.5(kW)TL273?20PL?10 an electric power generating plant has a rated output of 100MW. The boiler of the plant

operates at 300℃. The condenser operates at 40℃

(a) at what rate (joules per hour) must heat be supplied to the boiler?

(b) The condenser is cooled by water, which may under go a temperature rise of no more than

10℃. What volume of cooling water in cubic meters per hour, is require to operate the plant?

(c) The boiler tempeture is to be raised to 540℃,but the condensed temperature and electric

output will remain the same. Will the cooling water requirement be increased, decreased, or remain the same?

Data heat capacity 4.184, density 1g/cm3 (2.10)

(a)PH?TH300?2738P?10TH?TL300?408(b)

Solution: ?2.2?10(W)QL?4.3?1011(J)V?10?10?4.184?QLV?1.03?104(m3)6

QH?PHt?7.9?1011(J)(c)PH?TH540?2738P?10TH?TL540?40

?1.626?108(W)no11 (a) Heat engines convert heat that is available at different temperature to work. They have been

several proposals to generate electricity y using a heat engine that operate on the temperature differences available at different depths in the oceans. Assume that surface water is at 20℃, that water at a great depth is at 4℃, and that both may be considered to be infinite in extent. How many joules of electrical energy may be generated for each joule of energy absorbed from surface water? (b) the hydroelectric generation of electricity use the drop height of water as the energy source. in a particular region the level of river drops from 100m above sea level to 70m above the sea level . what fraction of the potential energy change between those two levels may be converted into electrical energy? how much electrical energy ,in kilowatt-hours, may be generated per cubic meter of water that undergoes such a drop? (2.11)

(a)W?Solution:

TH?TL20?4QH?1?0.055(J)TH20?273mg?h(b)P??3600?1.06?106(kW/h)1000

12 a sports facility has both an ice rink and a swimming pool. to keep the ice frozen during the summer requires the removal form the rink of 105 KJ of thermal energy per hour. It has been suggested that this task be performed by a thermodynamic machine, which would be use the swimming pool as the high temperature reservoir. The ice in the rink is to be maintain at a temperature of –15℃, and the swimming pool operates at 20℃, (a) what is the theoretical minimum power, in kilowatts, required to run the machine? (b) how much heat , in joule per hour , would be supplied t the pool by this machine? (2.12)

(a)P?Solution:

TH?TL20?155PL?10/3600?3.77(kW)TL273?15273?205(b)QH?10?1.14?105(kJ)273?15

13

(a)2Al?N2?2AlN(b)?H?152940(cal/mol)solution:(c)?S?4.82?2?6.77?2?45.77??49.67(cal/molK)(d)?H?152940(cal/mol)?S?4.82?2?6.77?2?45.77?8.314ln10??68.81(cal/molK) 14

40CP,ICEC?Hm?S?m(?dT???P,WATERdT)?10TTmT00

solution:?(2.1ln273336273?40??4.184ln)12000 263273273?22574(J/K)

15

W?TH?TL300?77QL?1000?2896(J)TL77

W2?70428(J)

16

W?W? 17

TH?TL300?4.2QL?83.3?5866.7(J)TL4.2TH?TL300?4.2QH?(83.3?1.5?8.314(300?4.2))?3719.4(J)TH300

(a)?T?0?U?OQ??W?n??pdV?1?8.314?298ln10?5704(J)P(b)?S?nRln0?1?8.314?ln10?19.1(J/K)P(c)Q?0(d)yes

18

500?60?TH?TL20?0335m?335mTL273

m?1222(g)

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