A coal-fired cogeneration station provides electricity and heat to a community of 1000 homes.
(a) Each home requires a constant $2.5 \mathrm {~kW} _ { \mathrm { e } }$, and the generation of electricity is $40 \%$ efficient. If the cogeneration station maintains an output that is just sufficient to satisfy the electrical needs of the community, what is the available total power associated with the waste heat?
(b) From part (a), what is the daily coal requirement in $\mathrm { kg }$ ?
(c) Each house has a volume of $450 \mathrm {~m} ^ { 3 }$ and maintains an inside temperature of $18.3 ^ { \circ } \mathrm { C }$. If $90 \%$ of the waste heat is available for heating purposes, what is the minimum (constant) outside temperature for which the cogeneration station can fulfil the heating needs of the community while just satisfying the electrical needs? See Chapter 8 for additional information on heating requirements.

Question

Quizplus · Accepted Answer

The Answer is (a) The total thermal energy generated is

( 1000 ) imes ( 2.5 \mathrm {~kW} ) / ( 0.4 ) = 6.25 imes 10 ^ { 3 } \mathrm {~kW}

The waste heat is, therefore,

60 \%

of the total or

\left( 6.25 imes 10 ^ { 3 } \mathrm {~kW} ight) imes ( 0.6 ) = 3.75 imes 10 ^ { 3 } \mathrm {~kW}

(b) The total daily thermal energy generated is

\left( 6.25 imes 10 ^ { 6 } \mathrm {~W} ight) imes ( 24 \mathrm {~h} / \mathrm { d } ) imes ( 3600 \mathrm {~s} / \mathrm { h } ) = 5.4 imes 10 ^ { 11 } \mathrm {~J} / \mathrm { d } = 5.4 imes 10 ^ { 5 } \mathrm { MJ } / \mathrm { d }

For an energy content of

31 \mathrm { MJ } / \mathrm { kg }

for coal, the total daily coal requirement is

\left( 5.4 imes 10 ^ { 5 } \mathrm { MJ } / \mathrm { d } ight) / ( 31 \mathrm { MJ } / \mathrm { kg } ) = 1.74 imes 10 ^ { 4 } \mathrm {~kg} / \mathrm { d }

(c) From Chapter 8 the heating requirement is

67 \mathrm {~kJ} / \mathrm { m } ^ { 3 }

per degree day

\left( { } ^ { \circ } \mathrm { C } ight)

. For 1000 homes of

450 \mathrm {~m} ^ { 3 }

the total energy requirement is

( 1000 ) imes \left( 450 \mathrm {~m} ^ { 3 } ight) imes \left( 6.7 imes 10 ^ { 3 } \mathrm {~J} / \mathrm { m } ^ { 3 } ight.

per degree day

) = 3.02 imes 10 ^ { 10 } \mathrm {~J}

per degree day
The amount of waste heat that is available at

90 \%

recovery is

( 0.9 ) imes ( 3.75 imes 106 \mathrm {~W} ) = 3.375 imes 10 ^ { 6 } \mathrm {~W}

or a total thermal energy per day of

\left( 3.375 imes 10 ^ { 6 } \mathrm {~W} ight) imes ( 24 \mathrm {~h} / \mathrm { d } ) imes ( 3600 \mathrm {~s} / \mathrm { h } ) = 2.92 imes 10 ^ { 11 } \mathrm {~J}

This is sufficient to provide heat for

\left( 2.92 imes 10 ^ { 11 } \mathrm {~J} ight) / \left( 3.02 imes 10 ^ { 10 } \mathrm {~J} ight.

per degree day

) = 9.7

degree days
For an inside temperature of

18.3 { } ^ { \circ } \mathrm { C }

the minimum outside temperature would be

\left( 18.3 ^ { \circ } \mathrm { C } - 9.7 ^ { \circ } \mathrm { C } ight) = 8.6 ^ { \circ } \mathrm { C }

This is a relatively high temperature and indicates that this cogeneration approach probably provides an insufficient amount of heat when it satisfies the electricity needs of the community.

A Coal-Fired Cogeneration Station Provides Electricity and Heat to a Community