CALCU-1 A gas-turbine power plant operates on the

Solution by Steps

step 1

To find the isentropic efficiency of the compressor, we need to compare the actual work done by the compressor to the work that would be done in an isentropic process

step 2

The isentropic efficiency of the compressor is given by the formula

\eta_{\text{isentropic}} = \frac{W_{\text{actual}}}{W_{\text{isentropic}}}

. However, since we are given the isentropic efficiency of the turbine, we cannot directly calculate the isentropic efficiency of the compressor without additional information

[question (i)] Answer

The isentropic efficiency of the compressor cannot be determined with the given information.

Key Concept

Isentropic Efficiency of Compressor

Explanation

The isentropic efficiency of the compressor compares the actual work to the work done in an ideal isentropic process. It requires knowledge of both actual and isentropic work or temperatures.

step 1

To determine the air-fuel ratio in the combustion chamber, we use the energy balance equation which relates the mass flow rate of air, the heating value of the fuel, and the temperature rise of the air

step 2

The energy added to the air can be calculated using the specific heat capacity of air and the temperature rise,

Q_{\text{added}} = \dot{m}_{\text{air}} \cdot C_{p,\text{air}} \cdot (T_{\text{out}} - T_{\text{in}})

step 3

The energy provided by the fuel is the product of the fuel's heating value and the mass flow rate of the fuel,

Q_{\text{fuel}} = \dot{m}_{\text{fuel}} \cdot HV_{\text{fuel}}

step 4

Equating the energy added to the air and the energy provided by the fuel, we get

\dot{m}_{\text{air}} \cdot C_{p,\text{air}} \cdot (T_{\text{out}} - T_{\text{in}}) = \dot{m}_{\text{fuel}} \cdot HV_{\text{fuel}}

step 5

Rearranging the equation to solve for the air-fuel ratio (AFR), we get

AFR = \frac{\dot{m}_{\text{air}}}{\dot{m}_{\text{fuel}}} = \frac{HV_{\text{fuel}}}{C_{p,\text{air}} \cdot (T_{\text{out}} - T_{\text{in}})}

step 6

Substituting the given values,

AFR = \frac{42000 \text{ kJ/kg}}{1.005 \text{ kJ/kgK} \cdot (871+273 - 400-273) \text{ K}}

step 7

Calculating the AFR, we find

AFR \approx 15.1

[question (ii)] Answer

The air-fuel ratio in the combustion chamber is approximately 15.1.

Key Concept

Air-Fuel Ratio

Explanation

The air-fuel ratio is calculated using the energy balance between the energy added to the air and the energy provided by the fuel, considering the specific heat capacity of air and the heating value of the fuel.

step 1

To determine the effectiveness of the regenerator, we use the formula

\epsilon = \frac{T_{\text{out, hot}} - T_{\text{out, cold}}}{T_{\text{in, hot}} - T_{\text{in, cold}}}

step 2

The temperatures are known from the problem statement:

T_{\text{out, hot}}

is the temperature of the gases leaving the turbine,

T_{\text{out, cold}}

is the temperature of the air after being heated in the regenerator,

T_{\text{in, hot}}

is the temperature of the combustion gases entering the turbine, and

T_{\text{in, cold}}

is the temperature of the air entering the compressor

step 3

Substituting the given temperatures,

\epsilon = \frac{871+273 - 400-273}{871+273 - 30+273}

step 4

Calculating the effectiveness, we find

\epsilon \approx 0.81

[question (iii)] Answer

The effectiveness of the regenerator is approximately 0.81.

Key Concept

Effectiveness of Regenerator

Explanation

The effectiveness of the regenerator is the ratio of the actual heat transfer to the maximum possible heat transfer, calculated using the temperatures of the hot and cold streams entering and leaving the regenerator.

for part 3, to determine the effectiveness, the formua need to include cp, and m air and m fuel

Solution by Steps

step 1

To determine the effectiveness of the regenerator, we need to use the formula that relates the heat transferred to the specific heat capacity and the mass flow rates of air and fuel

step 2

The effectiveness (ε) of the regenerator is defined as the ratio of actual heat transfer to the maximum possible heat transfer

step 3

The formula for effectiveness is:

ε = \frac{Q_{actual}}{Q_{max}}

, where

Q_{actual}

is the actual heat transferred and

Q_{max}

is the maximum possible heat transfer

step 4

The actual heat transferred can be calculated using the specific heat capacity of air (

c_p

), the mass flow rate of air (

m_{air}

), and the temperature difference across the regenerator

step 5

The maximum possible heat transfer is determined by the minimum of the hot and cold fluid heat capacities, which in this case is the product of

c_p

and

m_{air}

, since the air is both the hot and cold fluid in a regenerative cycle

step 6

The formula for actual heat transfer is:

Q_{actual} = c_p \cdot m_{air} \cdot (T_{out} - T_{in})

, where

T_{out}

and

T_{in}

are the outlet and inlet temperatures of the air, respectively

step 7

The formula for maximum possible heat transfer, assuming the air has a larger heat capacity than the fuel, is:

Q_{max} = c_p \cdot m_{air} \cdot (T_{hot} - T_{cold})

, where

T_{hot}

and

T_{cold}

are the temperatures of the hot and cold streams entering the regenerator

step 8

Finally, we can calculate the effectiveness by substituting

Q_{actual}

and

Q_{max}

into the effectiveness formula

Answer

The effectiveness of the regenerator is calculated using the formula

ε = \frac{c_p \cdot m_{air} \cdot (T_{out} - T_{in})}{c_p \cdot m_{air} \cdot (T_{hot} - T_{cold})}

.

Key Concept

Effectiveness of a regenerator

Explanation

The effectiveness of a regenerator is a measure of how well it transfers heat from the hot to the cold fluid relative to the maximum possible heat transfer, taking into account the specific heat capacity and mass flow rates of the fluids involved.

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