Consider a 2.4-kW hooded electric open burner in an area where the unit costs of electricity and...

Question:

Consider a 2.4-kW hooded electric open burner in an area where the unit costs of electricity and natural gas are $0.10/kWh and $1.20/therm (1 therm = 105,500 kJ), respectively. The efficiency of open burners can be taken to be 73 percent for electric burners and 38 percent for gas burners. Determine the rate of energy consumption and the unit cost of utilized energy for both electric and gas burners

Electric Heater:

An electric heater is a machine that is used to produce heat energy by converting electrical energy. In an electric heater, the material having higher resistance is used, which restricts the flow of electrons passing through it and generates heat.

Answer and Explanation:


Given Data


  • Energy input, {eq}{\dot Q_{in}} = 24\;{\rm{kW}} {/eq}.
  • Cost of energy for electric burner, {eq}{C_{EB}} = \ $0.10/{\rm{kWh}} {/eq}.
  • Cost of energy for gas burner, {eq}{C_{NG}} = \ $1.2/{\rm{therm}} {/eq}.
  • Efficiency of electric burner, {eq}{\eta _{EB}} = 73\% {/eq}.
  • Efficiency of gas burner, {eq}{\eta _G} = 38\% {/eq} .


The expression for rate of energy consumption in electric burner is,

{eq}{\dot Q_{cons}} = {\dot Q_{in}} \times {\eta _{EB}} {/eq}


Substituting the values in above expression.

{eq}\begin{align*} {{\dot Q}_{cons}} &= 24 \times 0.73\\ {{\dot Q}_{cons}} &= 17.52\;{\rm{kW}} \end{align*} {/eq}


Thus, the rate of energy consumption in electric burner is {eq}17.52\;{\rm{kW}} {/eq}.


Calculating unit cost of utilized energy in electric burner,

{eq}UC = \dfrac{{{C_{EB}}}}{{{\eta _{EB}}}} {/eq}


Substituting the values in above expression.

{eq}\begin{align*} UC &= \dfrac{{0.10}}{{0.73}}\\ UC &= \ $0.1369/{\rm{kWh}} \end{align*} {/eq}


Thus, the unit cost of utilized energy in electric burner is {eq}\ $0.1369/{\rm{kWh}} {/eq}.

The expression for rate of energy input in gas burner is,

{eq}{\dot Q_{cons,G}} = \dfrac{{{{\dot Q}_{cons}}}}{{{\eta _{GB}}}} {/eq}


Substituting the values in above expression.

{eq}\begin{align*} {{\dot Q}_{cons,G}} &= \dfrac{{17.52}}{{0.38}}\\ {{\dot Q}_{cons,G}} &= 46.105\;{\rm{kW}} \end{align*} {/eq}


Thus, the rate of energy consumption in electric burner is {eq}46.105\;{\rm{kW}} {/eq}.


Now, calculating unit cost of utilized energy in gas burner.

{eq}UC = \dfrac{{{C_{NG}}}}{{{\eta _G}}} {/eq}


Substituting the values in above expression.

{eq}\begin{align*} UC &= \dfrac{{1.2}}{{0.38}}\\ UC &= \ $3.157/{\rm{therm}} \end{align*} {/eq}


Unit conversion

{eq}\begin{align*} UC &= \dfrac{{3.157}}{{105500 \times \dfrac{1}{{3600}}}}\\ UC &= \ $0.107/{\rm{kWh}} \end{align*} {/eq}


Thus, the unit cost of utilized energy in gas burner is {eq}\ $0.107/{\rm{kWh}} {/eq}.


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