# Thermodynamic Systems - Power rating and thermal efficiency

#### Lewishio

Homework Statement
Hi,

I have had a go at this question but have gone wrong somewhere, any ideas?

Thanks.

A coal fired steam plant takes in feed water at a temperature of 70°C and produces 15 tonnes of steam per hour at a pressure of 60 bar and temperature 400°C.
The fuel consumption rate is 1.5 tonnes per hour and the calorific value of the fuel is 40MJkgˉ¹.
Determine the power rating of the boiler and its thermal efficiency.
Homework Equations
Q in = E x V
Q out = McΔT
Efficiency = Q out / Q in x 100
A coal fired steam plant takes in feed water at a temperature of 70°C and produces 15 tonnes of steam per hour at a pressure of 60 bar and temperature 400°C.
The fuel consumption rate is 1.5 tonnes per hour and the calorific value of the fuel is 40MJkgˉ¹.
Determine the power rating of the boiler and its thermal efficiency.

Power rating:
Q in = E x V
40,000,000Jkg^-1 x 1500kg p/h / 3600 = 16,666,666,67 J/s

Q out = McΔT
15,000kg x 4.19 KJ/kg°c x (400°c – 70°c) / 3600 = 5,761.25 J/s

Thermal efficiency:
Efficiency = Q out / Q in x 100
5,761.25 J/s / 16,666,666.67 x 100 = 0.034%

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#### Chestermiller

Mentor
From your steam tables, what is the enthalpy of steam at 400 C and 60 Bar. What is the enthalpy of liquid water at 70 C and 60 bar?

• Lewishio

#### Lewishio

Steam Enthalpy at 400°c at 60 bar = 3176.28KJ/kg
Water Enthalpy at 70°c at 60 bar = 297.89KJ/kg

#### Chestermiller

Mentor
Steam Enthalpy at 400°c at 60 bar = 3176.28KJ/kg
Water Enthalpy at 70°c at 60 bar = 297.89KJ/kg
How is the heat addition rate related to the steam mass flow rate and the enthalpy change per unit mass?

• Lewishio

I'm not sure

#### Chestermiller

Mentor
I'm not sure
Have you learned about the open system (control volume) version of the first law of thermodynamics?

Incidentally, in your original post, you omitted the heat of vaporization. You are aware of that now, right?

• Lewishio

#### Lewishio

I will do some research on open systems for the first law of thermodynamics and I can now see I havent included the heat of vaporization which I believe is 2,260KJ/kg. Im not too sure which equations are to be used to figure this out.

#### Chestermiller

Mentor
I will do some research on open systems for the first law of thermodynamics and I can now see I havent included the heat of vaporization which I believe is 2,260KJ/kg. Im not too sure which equations are to be used to figure this out.
Once you understand the application of the open system version of the first law to this, you will. I can help answer your questions.

• Lewishio

#### Lewishio

Hi Chester,

Do I need to find the volume of the mass entering the system?
If so, there are 15 tonnes entering, so this would be 15,000L.
Would I then multiply this by the Water Enthalpy at 70°c at 60 bar which is 297.89KJ/kg?
This would be 4,468,350 KJ.
Would I then multiply the mass by the heat of vaporization which I believe is 2,260KJ/kg?
This would be 33,900,000 KJ.

would the sum of these be the energy entering the system?
4,468,350 + 33,900,000 = 38,368,350 KJ

#### Lewishio

Is this the correct formula:

Q - W + Σm in (Hi + V1^2/2 + gzi) - Σm exit (He + V1^2/2 + gze)

#### Chestermiller

Mentor
Is this the correct formula:

Q - W + Σm in (Hi + V1^2/2 + gzi) - Σm exit (He + V1^2/2 + gze)
I don't see an equal sign in the above "equation." But this is definitely the correct starting point. For the case of the single water stream passing through the boiler, what does this equation reduce to (assuming negligible change in kinetic and potential energy)?

#### Lewishio

Equation assuming negligible change in kinetic and potential energy:
Q - W + Σmi x Hi - Σme x He = 0

#### Lewishio

I’m not sure if this is correct but thought I’d have a go:

Q - W + Σmi x Hi - Σme x He = 0

Q (Heat transfer):

McΔT
15,000kg x 4.19KJ/kg°c x (400°c - 70°c) = 20,740,500 KJ°c p/h

W (Work done):
Consumption rate x calorific value
1,500kg x 40,000KJkg = 60,000,000 KJ p/h

Heat of vaporization:
Q = mL
15,000kg x 2,260KJ/kg = 33,900,000 KJ

Total work done: 60,000,000KJ p/h + 33,900,000 KJ = 93,900,000 KJ p/h

H (Enthalpy):
MH

Steam Enthalpy at 400°c at 60 bar = 3176.28KJ/kg
Water Enthalpy at 70°c at 60 bar = 297.89KJ/kg

15,000kg x 297.89KJ/kg = 4,468,350KJ
15,000kg x 3176.28KJ/kg = 47,644,200KJ

Q - W + Σmi x Hi - Σme x He = 0
20,740,500 KJ°c p/h - 93,900,000 KJ p/h + 4,468,350KJ - 47,644,200KJ = - 116,335,350KJ

- 116,335,350KJ + 116,335,350KJ = 0

Power (watts)
116,335,350 / 3600 = 32,315.375 KW

#### Chestermiller

Mentor
I’m not sure if this is correct but thought I’d have a go:

Q - W + Σmi x Hi - Σme x He = 0

Q (Heat transfer):

McΔT
15,000kg x 4.19KJ/kg°c x (400°c - 70°c) = 20,740,500 KJ°c p/h

W (Work done):
Consumption rate x calorific value
1,500kg x 40,000KJkg = 60,000,000 KJ p/h

Heat of vaporization:
Q = mL
15,000kg x 2,260KJ/kg = 33,900,000 KJ

Total work done: 60,000,000KJ p/h + 33,900,000 KJ = 93,900,000 KJ p/h

H (Enthalpy):
MH

Steam Enthalpy at 400°c at 60 bar = 3176.28KJ/kg
Water Enthalpy at 70°c at 60 bar = 297.89KJ/kg

15,000kg x 297.89KJ/kg = 4,468,350KJ
15,000kg x 3176.28KJ/kg = 47,644,200KJ

Q - W + Σmi x Hi - Σme x He = 0
20,740,500 KJ°c p/h - 93,900,000 KJ p/h + 4,468,350KJ - 47,644,200KJ = - 116,335,350KJ

- 116,335,350KJ + 116,335,350KJ = 0

Power (watts)
116,335,350 / 3600 = 32,315.375 KW
This is totally screwed up. See my next post.

#### Chestermiller

Mentor
Equation assuming negligible change in kinetic and potential energy:
Q - W + Σmi x Hi - Σme x He = 0
I'm going to write this equation in a little different way:
$$\dot{Q}-\dot{W}+\dot{m}_{in}h_{in}-\dot{m}_{out}h_{out}=0$$
where $\dot{Q}$ = rate of heat transfer to water inside boiler (J/sec)
$\dot{W}$ = rate at which water inside boiler does shaft work (J/sec)
$\dot{m}_{in}$ = rate at which water enters the boiler (kg/sec)
$\dot{m}_{out}$ = rate at which water (steam) exits the boiler (kg/sec)
$h_{in}$ = enthalpy per unit mass of water entering the boiler (J/kg)
$h_{out}$ = enthalpy per unit mass of water (steam) exiting the boiler (J/kg)

Do you really feel that the water and steam inside the boiler does shaft work? If so, what is the force and displacement? If not, then $\dot{W}=0$

In this problem what is the rate at which liquid water enters the boiler $\dot{m}_{in}$?

In this problem, what is the rate at which water (steam) exits the boiler $\dot{m}_{out}$?

From your steam tables, what is the enthalpy per unit mass of liquid water entering the boiler $h_{in}$?

From your steam tables, what is the enthalpy per unit mass of the water (steam) exiting the boiler $h_{out}$?

• Lewishio

#### Lewishio

Thanks for your help, how does this look:

In this problem what is the rate at which liquid water enters the boiler ˙minm˙in?
4.16kg/sec
In this problem, what is the rate at which water (steam) exits the boiler ˙moutm˙out?
4.16kg/sec
From your steam tables, what is the enthalpy per unit mass of liquid water entering the boiler hinhin?
Water Enthalpy at 70°c at 60 bar = 297,890J/kg
From your steam tables, what is the enthalpy per unit mass of the water (steam) exiting the boiler houthout?
Steam Enthalpy at 400°c at 60 bar = 3,176,280J/kg

#### Chestermiller

Mentor
Thanks for your help, how does this look:

In this problem what is the rate at which liquid water enters the boiler ˙minm˙in?
4.16kg/sec
In this problem, what is the rate at which water (steam) exits the boiler ˙moutm˙out?
4.16kg/sec
From your steam tables, what is the enthalpy per unit mass of liquid water entering the boiler hinhin?
Water Enthalpy at 70°c at 60 bar = 297,890J/kg
From your steam tables, what is the enthalpy per unit mass of the water (steam) exiting the boiler houthout?
Steam Enthalpy at 400°c at 60 bar = 3,176,280J/kg
These numbers look about right. So, what is the heat load $\dot{Q}$ in J/sec?

• Lewishio

#### Lewishio

Would this be:
4.16kg/sec x 297,890J/kg = 1,239,222.4J/sec

#### Chestermiller

Mentor
Would this be:
4.16kg/sec x 297,890J/kg = 1,239,222.4J/sec
From the equation in post #15, if $\dot{m}_{in}=\dot{m}_{out}=\dot{m}=4.17\ kg/sec$, then $$\dot{Q}=\dot{m}(h_{out}-h_{in})$$What does this give you?

• Lewishio

#### Lewishio

4.17kg/sec x (3,176,280J/kg - 297,890J/kg) = 12,002,886.3J/sec

#### Chestermiller

Mentor
4.17kg/sec x (3,176,280J/kg - 297,890J/kg) = 12,002,886.3J/sec
Good. So what’s the efficiency?

• Lewishio

#### Lewishio

Thanks Chester, I can't seem to get the right answer for this:

Efficiency = Q out / Q in x 100

(4.17 x 3,176,280) ÷ (4.17 x 297,890) x 100 = 1066.26%

#### Chestermiller

Mentor
Thanks Chester, I can't seem to get the right answer for this:

Efficiency = Q out / Q in x 100

(4.17 x 3,176,280) ÷ (4.17 x 297,890) x 100 = 1066.26%
This is incorrect. The heating efficiency is defined as the heat that goes into generating steam divided by the energy available from burning the fuel (times 100)

• Lewishio

#### Lewishio

1,500kg/hour x 40,000,000Jkg / 3600 = 16,666,666.67 J/sec

12,002,886,3J/sec / 16,666,666.67J/sec x 100 = 72.01%

• Chestermiller

#### Lewishio

Thanks for your help, am I right in saying the power rating is 12MW?

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