Massflow of steam to be supplied to a drying cylinder

[lenilein]

TL;DR Summary

Massflow of steam to be supplied to a drying cylinder

Good afternoon,

I am trying to calculate the massflow of steam required in a cylinder used for paper drying but I think there is a bug in my calculation and I would love to get your help to find where the issue is!

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Saturated steam is continuously supplied to a cylinder. The steam condenses in the cylinder and the condensate is continuously removed from the cylinder as represented here:

A condensate layer along the cylinder inner surface and the cylinder shell separate the steam from the paper placed on the cylinder and the overall heat transfer coefficient is determined as:

U=1 / (1 / U condensate + 1 / U cylinder)

where H is the known heat transfer coefficient of the condensate, and since the cylinder has a large diameter:
U cylinder (W/m2.K) = thermal conductivity cylinder (W/m.K) / thickness cylinder shell (m)

The temperature of the paper web is calculated using an energy balance on the paper (taking into account the energy flow from steam to paper and convective heat transfer between the paper and the surrounding air)

According to physical laws of heat transfer:
Heat transferred from steam to paper (in Watt) = Area heat transfer (in m2) * U (in W/m2.K) * (T steam - T paper)

Which means that if the thermal conductivity of the cylinder shell is higher and other parameter values remain same, the resulting heat flow from steam to paper will be higher.

Now comes the tricky part: I am calculating the massflow of steam required in the cylinder as follows:

Massflow steam (kg/s) = 0.001 * Heat transferred from steam to paper (W) / Enthalpy of steam condensation (in kJ/kg)

However, this means that for a higher thermal conductivity of the shell the massflow of steam required will be higher => from my understanding this shouldn’t be the case since improved heat transfer should lead to steam savings.
Is my formula for the steam massflow maybe wrong?

Many many thanks in advance!

 

[BvU]

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Your picture has disappeared !

For the formulas it would be nice if you included units. The 0.001 comes out of the blue !

 

[lenilein]

thank you for the comment! I updated my post :)

 

[jrmichler]

It is very difficult to calculate heat transfer to paper from basic principles. Heat transfer to paper is the result of four heat transfer coefficients - steam condensate layer, through the condensate layer, through the dryer can shell, and dryer can to paper. The heat transfer coefficient from dryer can to paper is a function of paper moisture content, paper properties, and paper pressure (from the dryer fabric) against the dryer can. Paper temperature and moisture vary through the thickness of the paper. Depending on the grade of paper, the limit to heat transfer on the we end could be picking or the condensate layer, while the limit to heat transfer on the dry end is typically the dryer to paper heat transfer coefficient.

I spent ten years working in a paper mill as a mechanical engineer. One of my projects was a complete paper machine steam and condensate system. The old system ran all dryers at maximum 50 PSIG steam, the new system had the last ten dryers on the dry end running 75 PSIG steam. I had complete data from the old system, had been to a seminar by the Johnson Corp in Three Rivers MI, did a very careful job calculating, and the actual steam consumption in those ten dryers was almost exactly twice what I had calculated.

I remember calculating that the higher thermal conductivity of cold rolled silver compared to cast iron in dryer cans would meet company ROI requirements for increased production. For some strange reason, they did not buy into that one.

In another job, I had a program that calculated heat transfer to paper. That program was state of the art, made pretty graphs, but still was not very good at predicting.

The best approach that I ever saw was to extrapolate and interpolate from existing machines running similar paper grades at similar speeds. And don't forget the pocket ventilation systems - better pocket ventilation removes more moisture which cools the paper which causes greater $\Delta T$ which increases the heat transfer.

Are you a member of TAPPI (https://www.tappi.org/)? If not, you should be because they have the best information available on heat transfer to paper. I'm not aware of any good books or summary papers on the subject. That's where a TAPPI membership can be very useful.

 

[Chestermiller]

Massflow steam (kg/s) = 0.001 * Heat transferred from steam to paper (W) / Enthalpy of steam condensation (in kJ/kg)

However, this means that for a higher thermal conductivity of the shell the massflow of steam required will be higher => from my understanding this shouldn’t be the case since improved heat transfer should lead to steam savings.
Is my formula for the steam massflow maybe wrong?

Many many thanks in advance!

Where does the 0.001 come from? What fraction of the steam actually condenses before it leaves the cylinder?

 

[russ_watters]

Where does the 0.001 come from? What fraction of the steam actually condenses before it leaves the cylinder?

When using a trap, all of it. But it's not a given, it's a result.

...Pretty sure though the .001 is just the conversion of J to kJ.

 

[russ_watters]

I am trying to calculate the massflow of steam required in a cylinder used for paper drying but I think there is a bug in my calculation and I would love to get your help to find where the issue is!

...

​

However, this means that for a higher thermal conductivity of the shell the massflow of steam required will be higher => from my understanding this shouldn’t be the case since improved heat transfer should lead to steam savings.
Is my formula for the steam massflow maybe wrong?

I don't see that you've provided any description of your process at all. What you've calculated is the heating capacity of your system, not the heating requirement of your process. So yes, of course if you improve your system's heat transfer capability, the performance capability will increase (it's a tautology). But do you need it?

How much paper are you drying? What are its properties/requirements (size, mass, how wet is it, how fast is it passing over the drum?)? You seem to have calculated how much heat you can transfer to your paper, but you haven't calculated if that will actually dry the paper!

 

[lenilein]

First of all, THANK YOU for these many replies!
It is quite overwhelming to get so much support in a short time and to even read contributions from paper experts :)

Here my answers to your suggestions and comments:
- @Chestermiller : as @russ_watters rightly guessed the .001 corresponds to the conversion factor from W to kW (kJ/s).

- @Chestermiller: while writing the problem description I tried hard to simplify the problem as much as possible in order to keep the focus on my actual question. But you are right, in reality there is a share of so-called blow-through steam, which in the case I analyze is equal to 15%. Which means that only 85% of the steam condenses. In the case I described above I considered 100% condensation for the sake of simplicity but the formula can easily be adapted when considering the blow-through steam.

- @russ_watters :

What you've calculated is the heating capacity of your system, not the heating requirement of your process.

This is precisely the doubt that I have about the formula!

In my model the temperature and humidity profiles of the paper are calculated in a first step by integrating along the full length of the paper machine a differential equation system (dTp/dl and dxp/dl) which takes into account the conductive and convective heat and mass transfer processes occurring between the paper web and the surrounding mediums.

Then, as a consequence, I calculated the energy transferred with the obtained paper web temperature at each cylinder and thought it should be equal to the energy to be supplied to each cylinder but I have some strong doubts since I don't get the results I expect when simulating different cases (see case description in my answer to jrmichler below)

An alternative formula that I thought of is:
Massflow steam (kg/s) = Q (kW) / Enthalpy of steam condensation (in kJ/kg)

Q (kW) = G*v* w* ( ( cpw * xp final + cpf ) * Tp final - ( cpw * xp ini + cpf ) * Tp ini + ( xp ini - xp final) * Lvo) );

Where:
G: dry grammage of paper (kg d.s/m2)
v: speed of paper machine (m/s)
w: width of paper web (m)
cpw: specific heat capacity of water (kJ/kg water.K)
cpf: specific heat capacity of dry fiber (kJ/kg fiber.K)
Tp ini/final: Temperature of the paper web (and the water in it) at the beginning and end of the considered cylinder (°C)
xp ini/final: moisture content at the beginning and end of the considered cylinder (kg water/ kg d.s.)
Lvo: Latent heat of vaporization of water at 0°C => 2501 kJ / kg water

However, I see several issues in using that formula:

- in reality, the paper doesn't cover the whole cylinder surface so the energy that is transmitted to the air contacting the cylinder in those areas isn't included in Q
=> I thought of multiplying the Q by A = Area cylinder tot / Area cylinder covered with paper
but I'm not sure about the use of this A factor, since the air isn't at the same temperature as the paper web, and consequently I see no reason why the heat transferred through say 1m2 of non-covered cylinder should be equal to the heat transferred through 1m2 of covered cylinder.

- the evolution of temperature and humidity in the paper web depends not only on the heat transferred from the cylinder but also on the convective mass and heat transfer occurring with the surrounding air. Although it is clear that the convective heating through air is very marginal here, compared to contact heating with the cylinders, I don't know if my equation for Q is not too much of a simplification.

- @jrmichler: if I had talked with you about this 4 years ago, I might have chosen another phD topic :D
But it is too late now and I actually quite enjoy the challenge ... despite the doubts I go through on certain days!

The differential equation system for the variation in temperature and humidity of the paper web along an infinitesimal element of length (in machine direction) takes into account the four heat transfer coefficients that you mentioned, and the humidity and temperature of pocket ventilation air.

Also, I applied the model to literature data sets when humidity and temperature of the web were measured a many locations in the paper machine, and after calibrating one (maximum two) of the heat transfer coefficients which are hard to measure I got very good fits between the calculated and measured values (less than 5% error for both the humidity and temperature profiles).
So I thought that if I have come that far, determining the massflow of required steam shouldn't be too difficult...

Funnily the case you mentioned on replacing the iron dryers with silver is very similar to the reason of my post!
Right now there is a trend of replacing iron cast dryers with steel dryers because of their higher thermal conductivity and the lower thickness of the shell. In my work, I am trying to assess how much steam can be saved at a paper machine (or how many dryers removed) when doing so.
My issue is that, on one hand I do obtain, as expected, that the paper web is overdried when using the same pressure levels with steel cylinders, and therefore some cylinders can be removed. On the other hand, after virtually removing those cylinders from my calculation, I still obtain higher overall required steam massflow for the paper machine with steel dryers because of the additional amounts of steam calculated for higher thermal conductivity/lower thickness!

I have also heard from certain industrials (e.g. manufacturers of steam condensate systems) that they use extrapolations to determine the steam consumption. However, I also know that paper machine manufacturers do use models similar to mine when they design new equipments and/or calculate the effects of energy efficiency measures. Athough I'm not 100% sure if their models also include the massflow of steams to be supplied to the cylinder.

Yes I am a member of TAPPI and have read their technical papers on measures to improve the heat transfer coefficient of condensate etc. However it hasn't been of great help for setting up the physical equations.