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.

 

[ione]

The formula is ok, but you have to consider that specific heat is not constant in the temperature range between the saturation temperature and the actual (superheated steam) temperature. So using the value which corresponds to the superheated steam condition leads to an error (specific heat decreases as superheat degree increases).

 

[rkwolf]

I can see that being an issue, so if I just use T_superheated in shomate's equation, it given me Cp value for that temperature but as if the temperature I used is T_sat. Then how do I actually find Cp for superheated steam, knowing it is superheated steam and the temperature? Is their a correction factor? Or is it shomate's equation Cp plus something?

 

[rkwolf]

**there

 

[ione]

Think of the way specific heat is defined. It is the amount of heat per unit mass required to raise the temperature by one degree Celsius. So the problem is not with Shomate equation, the problem is with the assumption of considering the specific heat as a constant over the range of temperature you're dealing with (you are working with approx 222 °F of superheat, that is approx 123 °C).
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.

 

[hacksaw]

what reference temperature are you using for your enthalpy calulation?

 

[rkwolf]

I've been working on this for a few hours and I still can't figure it out. Here is what I got using shomate equation for Cp over the range of added superheat from T_sat=526K (488F) to T_super=650K (710F) with 4 degree intervals for the total goal of superheat used of 124K. In order to get my value to agree with superheated steam table, I should get 50.86 kJ/(kmol*K), so I am still definetily missing something. Thank you for your help.


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

 

[hacksaw]

the equation uses Abs T in Deg K not the superheat

 

[rkwolf]

That's what I did in the very beginning. Evaluating shomate equation at T=650K, I got:

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.

 

[IRstuff]

I don't get how this equation accounts for pressure if there's no pressure dependency in the equation.

TTFN

FAQ731-376
Chinese prisoner wins Nobel Peace Prize

 

[rkwolf]

Is there another way to find enthalpy of superheated steam besides the steam table?

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?

 

[ione]

After a double check with webbook NIST, I’ve seen the Shomate equation gives good results at atmospheric pressure (ideal gas behaviour). Your superheated steam is at 610 psia and Shomate equation definitely gives wrong results. I suggest you to go webbook NIST (

http://webbook.nist.gov/chemistry/fluid/

) in order to get Cp vs temperature at 610 psia for water and then try to extrapolate a best fit curve in your temperature range.
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.

 

[PaoloPemi]

if I uderstand your question you need to calculate water / steam enthalpy in a range of temperatures/pressures, if that range is limited you can possibly define a set of simplified correlations (for example by fitting parameters with data regression from steam tables, see IAPWS95/97) , as alternative I would suggest to create a large table of values (for example the enthalpy (of saturated fluid) at regular intervals of temperature plus several points (iso pressure) for temperatures above and below, then interpolate.
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

 

[hacksaw]

the op is using the zero pressure specific heat,

@rkwolf

what saturation enthalpy is your table showing?

 

[rkwolf]

@PaoloPemi

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)

 

[hacksaw]

The Shomate equation you are using

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,

 

[rkwolf]

Thanks everybody for your help. I think I will try both methods and see what I get.

@ 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

 

[racookpe1978]

1) What range of pressures will you be needing to work? hat will reduce the size of the data fields.

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?

 

[rkwolf]

@ racookpe1978

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.

 

[pmover]

rkwolf,

perhaps this website may be of help to you . . .

http://www.x-eng.com/Download_XSteam.htm

good luck!
-pmover