Calculate the amount of natural gas required to heat the water in a heater.
Answer to Problem 41P
The amount of natural gas required to heat the water in a heater is 72 ft3day.
Explanation of Solution
Given data:
Volume of water used per day is 80 gallons,
Initial temperature of water is, Tinitial=55 °F,
Final temperature of water is, Tfinal=140 °F,
Efficiency of heater is 80 %,
From Table 11.13 in the textbook, the specific heat of water is, c=4180 Jkg⋅K,
Standard density of water is 1000 kgm3.
Formula used:
The relationship between degree Fahrenheit and degree Celsius is,
T(°C)=59[T(°F)−32] (1)
Here,
T(°C) is the temperature in degree Celsius,
T(°F) is the temperature in degree Fahrenheit.
The formula to find thermal energy required to heat the water is,
Ethermal=mc(Tfinal−Tinitial) (2)
Here,
m is the mass of the water,
c is the specific heat of water,
Tinitial is the initial temperature of water,
Tfinal is the final temperature of water.
Calculation:
Convert the unit of volume of the water,
Volume=80 gallonsday [∴1 gallon=3.7854×10−3 m3]=(80 gallonsday)(3.7854×10−3 m31 gallon)=(80×3.7854×10−3) m3dayVolume=0.3028 m3day
Substitute 55 °F for T(°F) in equation (1) to get the initial temperature in degree Celsius,
Tinitial(°C)=59[55−32]=12.77 °C
Substitute 140 °F for T(°F) in equation (1) to get the final temperature in degree Celsius,
Tfinal(°C)=59[140−32]=60 °C
The expression to find the mass of the water is,
Mass=Volume×density=(0.3028 m3day)(1000 kgm3)Mass=302.8 kgday
Substitute 302.8 kgday for m, 4180 Jkg⋅K for c, 12.77 °C for Tinitial, and 60 °C for Tfinal in equation (2) to find Ethermal,
Ethermal=(302.8 kgday)(4180 Jkg⋅K)(60 °C−12.77 °C) [∴ °C=K]=(302.8 kgday)(4180 Jkg⋅K)(60−12.77) K
Ethermal=59.7792×106 Jday
The unit conversion on above result is,
Ethermal=59.7792×106 Jday [∴1 day=24 h]=59.7792×106 (Jday×1 day24 h) [∴1 h=3600 s]=(2.491×106 Jh)(1 h3600 s) [∴1 Js=1 W]Ethermal=693 W
Do the unit conversion on above result,
Ethermal=693 W [∴1 W=3.412 Btuh]=(693 W)(3.412 Btuh1 W)Ethermal=2364.6 Btuh
At standard 77 °F temperature, the heating value of methane is 978 Btuft3. Since the major composition of natural gas is methane, consider the heating value of natural gas is 978 Btuft3, as the range of temperature given as 55 °F to 140 °F considered for 77 °F.
Then the amount of natural gas required to heat the water in heater can be determined from the below expression:
Volume of gas=EthermalEfficiency×Heating value of gas=2364.6 Btuh0.8(978 Btuft3) [∴1 day=24 h]=(3 ft3h)(24 h1 day)Volume of gas=72 ft3day
Therefore, the amount of natural gas required to heat the water in a heater is 72 ft3day.
Conclusion:
Hence, the amount of natural gas required to heat the water in a heater is 72 ft3day.
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Chapter 11 Solutions
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