Specific Heat Constant for Superheated Steam 1

[rkwolf]

Hell all,

The company I recently started working for, asked me to design boiler efficiency program that can monitor efficiency in real time using instantenious (almost) inputs from several flow meters as well as temperature and pressure transducers. The efficiency equation itself is not complicated and I could evaluate efficiency given the flow rates, pressures, temperatures and steam tables. The tricky thing is since this is supposed to be a program able to calculate efficiency at any given time, the use of table is out of the question, meaning I have to come up with equations describing each pressure-temperatue relation.
I was doing ok working on it until I ran into a problem. If the outlet steam is superheated, how do I evaluate it's enthalpy. My plan was to use

h(superheated) = h(sat. vapor @ T_sat for given boiler P) + Cp*(T_steam - T_sat)

The issue came up when evaluating specific heat constant for superheated steam. I used Shomate's Equation:

Cp° = A + B*t + C*t2 + D*t3 + E/t2

where t=(temperature in Kelvin)/1000
A=30.09200
B=6.832514
C=6.793435
D=-2.534480
E=0.082139

However, checking my result against the steam tables, showed inaccuracy. I will provide an example below:


P(operating pressure)=610 psia;
T_steam= 710 F
T_sat=488.1 F (@ operating pressure)
h_g = 1203 Btu/lb
T_delta= T_steam-T_sat=710-488.1=221.9 F

Using Cp equation from above,
Cp=36.9017 kJ/(kmol*K) = 2.053 kJ/(kg*K) = 0.490 Btu/(lb*F)

Plugging into the enthalpy equation, I got:

h(superheated steam) = 1315 Btu/lb

However, the steam table give the result of 1355.7 Btu/lb

I was hoping someone can explain to me where the difference comes from? Is shomate's equation not valid for superheated steam? If so, is the another formula I could use? Or maybe I am missing something else entirely? Any help would be appriciated.

 

[hacksaw]

The Shomate equation gives you the zero pressure specific heat. While itt agrees pretty well with Eqn of State at low pressures, there is a considerable departure as you increase the steam pressure.

As you have started to do, the zero pressure correlation it can be used to estimate the enthalpy of superheated steam, given the saturation enthalpy. However, with a temperature dependent cp you have to integrate the specific heat from the saturation temperature to the superheat condition. A simple product of cp x delta T won't do.

Since pressure effects have been excluded from the specific heat estimate it will have significant error.

For example,

H(710F) = HSat+Integral[cp(t)dt] from 488 Deg F to 710 deg F

When you do this you get:

H(710 F)=1202+106.6 BTU/lbm= 1308.6 BTU/lbm (off by 48 BTU/lbm)

You can generate a specific correlation for the specific heat in the pressure/temperature range of interest and get much better results but you still have to integrate over the temperature interval.


For what its worth...

 

[ione]

rkwolf,

You can start with the spreadsheet suggested by pmover (check his post 14 Jun 11 21:21) and expand/modify it for your own requirements.

 

[SomptingGuy]

Can you not just integrate Cp(T).dT over the T range? The Cp(T) function looks fairly trivial.

- Steve

 

[rkwolf]

Thanks to everybody for their help. I ended up creating one big table of temperature and pressure (for superheated region), with values of enthalpy as independent variable. The way I was planning on doing it in my post from 14 june 14:28.

Afterwards, with some rearranging I was able to input it into DataFit and came up with algorithm to describe enthalpy as a function of two variables (pressure, temperature).

My error is about 0.3% from the table values, which for my purposes is good enough. Again, thanks to everybody for their help.