Specific Heat Constant for Superheated Steam
Specific Heat Constant for Superheated Steam
(OP)
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.
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.





RE: Specific Heat Constant for Superheated Steam
RE: Specific Heat Constant for Superheated Steam
RE: Specific Heat Constant for Superheated Steam
RE: Specific Heat Constant for Superheated Steam
Break your temperature range into intervals and compute the specific heat for each interval (you can do this with Shomate equation). The smaller the intervals the closer the result to the superheated steam table.
RE: Specific Heat Constant for Superheated Steam
RE: Specific Heat Constant for Superheated Steam
T Cp
(K) kJ/(kmol*K)
5 3315.69
10 851.55
15 395.26
20 235.58
25 161.69
30 121.57
35 97.39
40 81.71
45 70.98
50 63.31
55 57.64
60 53.34
65 50.01
70 47.37
75 45.24
80 43.52
85 42.09
90 40.90
95 39.90
100 39.05
105 38.33
110 37.71
115 37.17
120 36.71
125 36.30
RE: Specific Heat Constant for Superheated Steam
RE: Specific Heat Constant for Superheated Steam
Cp=36.9017 kJ/(kmol*K) = 2.053 kJ/(kg*K) = 0.490 Btu/(lb*F)
That is a legit value, however it applies for T=T_sat. The value for Cp at T_super=650K, should be around 0.675 Btu/(lb*F). That is where I got stuck.
RE: Specific Heat Constant for Superheated Steam
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RE: Specific Heat Constant for Superheated Steam
Using
h_super=h_sat + Cp*(T_super-T_sat)
would have been really nice and straighforward, but it is no use if I can't figure out the Cp.
Like I said in eariler post, the whole point of this is to be able to write a program able to determine efficiency in real time and respond to changes/disturbances. I can't program all the superheated steam tables for different pressures in there. Is my only option to determine equation
h_super=A*T_super+B
for a number of common boiler pressures and then let the software interpolate between?
RE: Specific Heat Constant for Superheated Steam
Once again you have to use a mean value as in your temperature range Cp varies from 0.98 Btu/(lb °F) at 610 psia and 488.09 °F to 0.583 Btu/(lb °F) at 610 psia and 710 °F.
RE: Specific Heat Constant for Superheated Steam
Accessing a 2D table is a very fast process and you can obtain reasonably accurate values.
For very accurate values create a very large table or use a complex model as IAPWS95 which requires a iterative solution , not very fast, to get a idea of timing see for example the libraries at prode.com
RE: Specific Heat Constant for Superheated Steam
@rkwolf
what saturation enthalpy is your table showing?
RE: Specific Heat Constant for Superheated Steam
it's worth a shot. I'll try to come up with one big table listing superheated enthalpies for different pressures for given tempatures, like this:
P
T 200 psia 275psia 300 psia .....
------------------------------------------
200 h h h
250 h h ....
300 h h
350 h ...
.... .....
@hacksaw
for P=610 psia, T_sat=488 with
h_f=473, h_fg=729, h_g=1202
h_super(@T_super=710)=1355 (need)
RE: Specific Heat Constant for Superheated Steam
uses a reference temperature of 298.15 DegK
Most Steam table use 273.15 Deg K, so is you add the Cp of liquid water for a delta T of 25 Deg K, to you Cp calculation, you'll get better agreement,
may not be as good as you planned interpolation table, but it uses a lot less memory,
RE: Specific Heat Constant for Superheated Steam
@ PaoloMPemi
Table is alot of work, however, my boss said he has access to some data analysis software that should be able to pull out two-variable equation:
h=A*pressure + B*temperature (that's an example, obviously it will be alot more complicated than that)
I should be done with the table tomorrow, and we'll see what happens.
@ hacksaw
that's a good point, i will mess it with some more and let you know how it went
RE: Specific Heat Constant for Superheated Steam
I like the idea of a lookup table, interpolating between points of temperatures and pressures.
2) When your program leaves the "bounds" of the table, how will you "gracefully exit" so no control failure causes problems? Or, am I reading the intent wrong, and this not be a "control" program that is manipulating plant systems, but a "monitoring program" that only evaluates performance?
RE: Specific Heat Constant for Superheated Steam
1) The normal industrial boiler range ... whatever that is. My boss wants to be able to sell this to whoever has a mid to large size boiler, mostly in waste treatment industry. [fuel type: natural gas, oil and sewer gas]
2) This is just "monitoring program" as of right now. The intent is to just make sure the boiler is operating at the appropriate efficiency. This may change if the current set-up (what I am working on right now) works well and the goal may be expanded.
RE: Specific Heat Constant for Superheated Steam
perhaps this website may be of help to you . . .
http://www.x-eng.com/Download_XSteam.htm
good luck!
-pmover
RE: Specific Heat Constant for Superheated Steam
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...
RE: Specific Heat Constant for Superheated Steam
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.
RE: Specific Heat Constant for Superheated Steam
- Steve
RE: Specific Heat Constant for Superheated Steam
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.