# Mathematical Modeling of an evaporator for evaporating water

• vishnu123
In summary: There are a lot of them available online. For example, Luyben Process Modeling, Simulation and Control for Chemical Engineers.
vishnu123
I am trying to model an evaporator which evaporates water to steam in Matlab/simulink. Are there any governing equations i need to know and follow when modelling the evaporator mathematically? Please let me know.

Hi,
vishnu123 said:
Are there any governing equations i need to know
Yes there are, of course.
Where do you want to start ? Or do you want to hire someone to do the work for you ?

thank you for the reply. I need to start from the scratch and i need to model the evaporator in simulink. are there any references to find the governing equations.

Did it occur to you to e.g. google 'evaporator equations' ?

yes i tried but i need equations with respect to time and in a simple way.

Mass Transfer Operations, Treybel

vishnu123 said:
yes i tried but i need equations with respect to time and in a simple way.
What about a textbook, e.g. Luyben: Process Modeling, Simulation and Control for Chemical Engineers

i have tried but i didnt find anz information regarding evaporator.

Have you heard of things like mass balances, heat balances, etc ?
What is the curriculum you are following that gives you exercises without providing a basis ?

I have done with the energy balance equations. the issue which i am facing is that i just know only one information about the parameters for the evaporator. the input will be the water at room temperature, which has to be evaporated to steam and this water steam is my output. and i am dealing with molar flow rates and not mass flow rates. the molar flow rates will always remains constant through out the process because there is no mixing of any other products. as there is a phase change and the steam is the final output, i just wanted to know how such type of evaporator is modeled.

Good. so you have a molar balance. In addition you will need an enthalpy (heat) balance. Then some way to bring in that heat: a heat exchanger is the usual equipment.
Make a list of all variables involved and a list of equations and post them.
Next step is to decide which ones are given, which ones need to be assumed and which ones can then be calculated.

A drawing of the setup is also highly desirable and helpful !

Luyben spells it out. So do many other textbooks.

My second question in #9 is still open.

One reason you don't get a full answer directly here is related to PF culture (see rules and guidelines) . But I am convinced you are trying hard, so let's proceed.

thank you, i am from computational background student doing my master thesis. for my thesis i jus need to mathematically design an evaporator as a minor sub task in simulink. and its been a while since i went through heat and Refrigeration stuff, so i am bit confused. I will post the total equations what i have scribbled.

here are the equations please check and do the necessary corrections.

Working fluid energy balance:
M_f*(dh_f/dt) = m_f(h_fin-h_fout)+A_f*U*(T_w-T_fin);Wall energy balance:
M_w-C_pw*(dT_w/dt) = -A_f*U*(T_w-T_fin)-A_g*U*(T_w-T_gin)

Exhaust gas energy balance:
M_g-C_pgin*(dT_g/dt) = m_g(C_pgin*T_gin - C_pgout*T_gout)+A_g*U*(T_w-T_gin)

Where M_f, g = mass of fluid, gas
m_f,g = molar flow of fluid, gas
h_f = enthalpy of the fluid
A_f = area of the tube
U = heat transfer coefficient
T_w = boiling temperature of water
T_f = fluid temperature
M_w = mass of the wall
C_pw, g = specific heat of the wall, gas
A_g = area of the shell
T_g = temperature of the gas

Last edited:
vishnu123 said:
here are the equations please check and do the necessary corrections.

Working fluid energy balance:
$$M_f\,{dh_f\over dt} = m_f\left ( h_{f,in}-h_{f,out}\right )+A_f \,U\,\left (T_w-T_{f,in}\right )$$Wall energy balance:
$$M_w-C_{p,w}\,{dT_w\over dt} = -A_f\,U\,(T_w-T_{f,in})-A_g\,U\,(T_w-T_{g,in})$$

Exhaust gas energy balance:
$$M_g-C_{p,\;g,{in}}{dT_g\over dt} = m_g(C_{p,\;g,{in}}\,T_{g, in} - C_{p,\;g,out}\,T_{g,out})+A_g\,U\,(T_w-T_{g,in})$$

Where ...
(your post was hard to understand until I typeset the balances)

Several questions:
1. What are you describing ? Drawing ?
2. Units ?
3. What tube ? What shell ?
4. using M and m in a mixed mode is asking for errors
5. Do you want an exhaustive, detailed dynamic model or something a bit simpler to begin with ? Perhaps even just a steady state model ?
6. With 14 variables and 3 equations you still need to fix 11 variables before you can calculate anything. Which ones ? Either that, or you need more equations. See also point 5.
7. You imply all U are the same ?
8. Are you interested in the temperature of the wall ?
9. etc
I really recommend you search for a simple textbook. Preferably your own from 'a while' ago (> 40 years ?)

so, basically i am using a shell and tube evaporator. units are all in SI units and temperature in kelvin. and M referes to the mass of the fluid and m referes to mass flow rate. I am looking for a detailed dynamic model of the evaporator if the unknowns are too many i can go with the basic model. U referes to over all heat transfer coefficient, which can be calculated using the basic equation. wall can be referred as a outer dia of the tube. I think i have all the other details like specific heat, inlet temperature of the cold and the hot fluid. what i need is the outlet temperature of cold fluid(water) and it has to be in vapour state. i have gone through few books like Refrigeration and air condition by P.K Nag, R.S Kurmi. If you know any better books can you please suggest me. In the equations it is not substract its a product on left hand side of the equation

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vishnu123 said:
In the equations it is not substract its a product
Sorry for misinterpreting your minus sign . If you use the right-mouse button while hovering above a ##\LaTeX## typeset equation, you can see the input to render equations in PF.

Very useful to learn some ##\TeX## for PF and elsewhere later on (I learned mine in the eighties... )
(enclose in  for displayed math, centered in new paragraph; or enclose in ## for in-line math formulas).

vishnu123 said:
I am looking for a detailed dynamic model of the evaporator if the unknowns are too many i can go with the basic model.
My point is that it is much easier to develop a working simple model first and then evolve it into something more detailed. You are basically setting up and solving an N equations times N variables calculation. If N is twice as big, it is four tiems more difficult to get it to work and check that is is correct.
In that context:
vishnu123 said:
Refrigeration and air condition by P.K Nag, R.S Kurmi. If you know any better books
Don't know Nag and Kurmi, but I suspect a lot of detail. Much better if you can find something more introductory such as Luyben. I have 2nd ed (1989 -- so don't buy but borrow) and ch 3.7 'Single component vaporizer' taylormade for your exercise !

Alternative tip:
Split up your problem in compartments. First evaporate using a fixed ##Q## enthalpy flow in a steady state model and worry about utilities later.PS I quite like your exercise: it's been my work for over 30 years and I'm about to retire, leaving behind a lot more work on the same subject .

thank you so much for the help. now i have a clear view of the task, problem defination and way of approach. After quite research i think i will procede with electric evaporator instead of shell and tube, because its very simple and reliable for my task. I will proceed with electric evaporator now. I am very greatful to you for your time and help. Have a nice day. If i find any difficulties i hope i can contact you

BvU
hello @BvU
I have modeled the equation for electric evaporator. I have attached the file here. But i have a small doubt in it how can i find that there is a phase change? I mean from liquid water to vapour. To my knowledge if there is a change in enthalpy that means there is a phase change. Is it correct?Can you check the attached file and can let me know wether i can use this equation for evaporation of water or not. If not please guide me through this.

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Again, I wonder what you are modelling: if U = 0, as in the case of perfect insulation, you have less to worry about.

vishnu123 said:
To my knowledge if there is a change in enthalpy that means there is a phase change. Is it correct?
No. But the other way around: yes. If there is a change in phase, there is a change in enthalpy. A change, a jump, a discontinuity, whatever you want to call it. The enthalpy changes by ##h_{fg}## when the phase changes from liquid to gas at the boiling temperature.

A change in enthalpy also occurs with a change in temperature: ##c_p \equiv {dH\over dT} \Big |_P##
(liquid and vapour have different ##c_p##)

Table A.7 here is public, google 'PROPERTY TABLES AND CHARTS (SI UNITS)' for something better. Or use the steam tables once you understand what is going on in your setup.

vishnu123
BvU said:
Again, I wonder what you are modelling
The equation is suitable for an immersion heater with heat loss from the cup to the environment .

Please use ##\LaTeX## instead of png so the thing can be referenced and usefully processed otherwise.

I am sorry for the png. I will post a latex version from now on.
BvU said:
The equation is suitable for an immersion heater with heat loss from the cup to the environment

And yes, exactly I have modeled an immersion water heater. I got an idea in mean time that why can't i use an immersion water heater as evaporator? In precise both does the same work but when we use it as an evaporator there will be only a phase change from liquid to vapour. so if i did the enthalpy balance in this equation, I think i can use the same equation as an electric evaporator. Can you tell me your opinion on it.

Sorry i understood now. I need to model an electric boiler but not an electric heater if i want to convert water into steam.

Right !

vishnu123

## 1. What is mathematical modeling of an evaporator for evaporating water?

Mathematical modeling of an evaporator for evaporating water is the process of creating a mathematical representation of the physical system of an evaporator used to remove water from a liquid mixture. This model helps to predict the behavior and performance of the evaporator under different conditions.

## 2. Why is mathematical modeling important for evaporators?

Mathematical modeling is important for evaporators because it allows engineers and scientists to analyze and optimize the design and operation of the evaporator. This can lead to improved efficiency, reduced energy consumption, and cost savings.

## 3. What factors are considered in mathematical modeling of an evaporator?

The factors considered in mathematical modeling of an evaporator include the physical properties of the liquid mixture, heat transfer coefficients, mass transfer coefficients, and the design and operating conditions of the evaporator such as temperature, pressure, and flow rate.

## 4. How accurate are mathematical models of evaporators?

The accuracy of mathematical models of evaporators depends on the complexity of the model and the availability of accurate data for the system. Generally, the more detailed and accurate the model is, the more accurate the predictions will be. However, there may still be some discrepancies between the model and the actual system due to uncertainties and variations in real-world conditions.

## 5. Can mathematical models be used to optimize evaporator performance?

Yes, mathematical models can be used to optimize evaporator performance by simulating different operating conditions and designs to determine the most efficient and cost-effective configuration. This can help to reduce energy consumption, improve product quality, and increase overall efficiency of the evaporator.

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