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Thermodynamics topics

Example of high precision equation of state

Thermodynamics fundamentals - Part 2

Still working on thermodynamics problems....

 

We have the change in Gibbs free energy for the following reactions (every compound is a separate phase:

$Al_2O_3(s)+SiO_2(s)=Al_2SiO_5(s)$ $\Delta G^{\circ}=-8320-0.4T$ $J$

 

$3Al_2O_3(s) +2SiO_2(s)=Al_6Si_2O_{13}(s)$ $\Delta G^{\circ}=22770-31.8T$ $J$

a) Compute $\Delta G^{\circ}$ for the following reactions:

 

$3Al_2SiO_5=Al_6Si_2O_{13}+SiO_2$

 

$Al_6Si_2O_{13}=2Al_2SiO_5+Al_2O_3$

 

also compute the temperatures for which $\Delta G^{\circ}=0$

 

b) Plot the phase diagram for the system $SiO_2\cdot Al_2O_3$ between 45 and 65% mol percent $Al_2O_3$ at atmospheric pressure and the temperatures computed on a9)

c) For the reaction $3Al_2SiO_5=Al_6Si_2O_{13}+SiO_2$, $\Delta V>0$ ¿How would the coexistence temperature of these phases change if pressure increases?

The solution can be found on this Juypter notebook.

You can also check part one of this series here.

Thermodynamics fundamentals

 Recently I have been reading Principles of extractive metallurgy , I'm currently at the starting chapters, mostly thermodynamics review. I will create an entry for every chapter with a couple of solved problems. 

So for this week these are the exercises:

1) For the reaction $H_2\text{(g, 1 atm)}+\frac{1}{2}O_2\text{(g, 1 atm)}=H_2O\text{(l)}$, $\Delta H^{\circ}_{298}=-285.9$ kJ and $\Delta G^{\circ}_{298}=-238$ kJ.

a) Compute $\Delta S^{\circ}_{298}$

b) The reaction can proceed at 298 K and 1 atm in a reversible way, for example, in a fuel cell, or in a completely irreversible form, that is, by combustion with no other work besides volume. Compute for both cases the heat $q$ absorbed from the environment and, for case a), the work $w'$ done on the environment. Show that your results are in agreement with the second law of thermodynamics.

Now I will try using a jupyter notebook for storing the solutions, I think the possibility of combining text and code could be useful for problems that requires several calculations.

The solution can be found here.

 

 

 

 

 

 

 

 

 

 

Finding drying time from drying curve

This post shows how you can find the time that will take for a product to reduce its moisture content if you have a drying curve at hand.

Definition of drying rate

The drying rate $R$ is defined as:

$$R = -\frac{m_s}{A}\frac{dX}{dt}$$ Where $m_s$ is the mass of dry solid, $A$ is the area of heat/mass transfer and $X$ is humidity in mass of water per mass of dry solid. The rate also can be given in the form $R'=RA/m_s$. A drying rate curve shows $R$ as a function of $X$. To find the total drying time a simple integration is performed. But keep in mind that a drying curve is valid only for a specific equipment under specific operating conditions. 

Example

This example was taken from Seader's Book. Experimental data is provided for drying of rayon waste and it's required to estimate the drying time starting from $X=1$ to $X=0.10$.

Experimental data

A numerical integration using the average value of $R'$ between every pair of points gives a drying time equal to 5 [min]. The detailed calculation can be found on this file.

Calculation of adiabatic saturation temperature

We will review how to calculate the adiabatic saturation temperature. The definition of this quantity was given in a previous post.

Problem definition

Given a gas-vapor mixture, if the inlet temperature and humidity are known, find the adiabatic saturation temperature.  The following equation is used:

From the inlet conditions we know $T_{\text{air}}$, Y and C. The main problem is that the humidity at saturation $Y_{\text{sat}}$ and the enthalpy of vaporization $\lambda$ depend on $T_{\text{as}}$. So an iterative calculation needs to be employed. Fortunately it can be easily implemented in Excel.

Example

This example comes from Seader's book. Air with inlet temperature 140 [F] and 12.5% relative humidity of water enters an process. Find the adiabatic saturation temperature of this current.

To solve this we also need enthalpy of vaporization and vapor pressure of water, and specific heat of both air and water, all as functions of temperature. All the required correlations can be found on Perry's book.

After a few trials by hand, the result is found using solver, and is equal to 87 [F]. The detailed calculations are on this file.

The wet bulb temperature

I started to study drying of solids, so is a good opportunity to revisit this important concept. The following results were adapted mainly from Treybal's book.

The humidity

Absolute humidity (Y) is simply the ratio of water mass as vapor and the air mass, expressed here in terms of partial and total pressures. Based  on the equality of the molar and pressure ratio:

 

Where M are the respective molecular weights.

Measurement of wet bulb temperature

Imagine a wet cloth that is exposed to an air stream. If the air is not saturated, water will evaporate from the cloth. The energy required for evaporation comes initially from the rest of the water, decreasing its temperature. Now due to the difference in air and water temperatures, heat will flow from the air to the remaining water.

As more water evaporates the remaining water will keep cooling, increasing the heat transfer rate from the air, until an equilibrium is reached. At this point the heat flow from the air is just enough to sustain the water vaporization. The following relation is satisfied:

 

 Adiabatic cooling

Now, imagine another process, this time air flows over a liquid surface. Similar to the previous case, water evaporates, taking energy from the air in the form of heat.  If this process is carried without external heat flow, it is termed adiabatic.

Before, air temperature and humidity were assumed constant, as the cloth is small compared to the air flow. This time however, both the temperature and humidity of the air change appreciably. The air humidity increases until the air becomes saturated.

A heat balance for the process gives:

Where C is the heat capacity of the inlet humid air. This equation says that the heat lost by the air together with its starting humidity is equal to the heat needed to add the additional humidity. 

Relation between wet bulb temperature and adiabatic cooling

Now lets see the equations for both processes side by side:

It turns out that the following ratio, called the Lewis relation is approximately one for an air-water mixture. So the wet bulb temperature (Teq) is approximately equal to the adiabatic saturation temperature (Tas).

Solving a flow distribution problem

Over the past few weeks I had to study fluid mechanics again, so I thought that for this week post I could share an example of that.

Problem description

Given a network of pipes, where the length and diameter of each section is known, calculate the flow of water in the network (the Bernoulli boundary values must also be known, details later). Lets solve for the network and data that follows (the construction material is PVC).

Section	Length[m]	Diameter[m]
1	2800	0.25
2	1300	0.18
3	800	0.18
4	1300	0.13

Theory

The Bernoulli equation says that the energy difference between two nodes is due to the friction losses by fluid flow:

The friction losses (h) are proportional to the length of the section (L) and the adimensional head loss (J). Several models are available for J, for water the Hazen-Williams correlation is:

Where the coefficient C is specific for the material. There are four unknown flow rates (Q), we can write three Bernoulli differences between pairs of boundary nodes, and the last equation is the flow rate balance at the central node.

This kind of problems are solved easily using Excel's solver, giving as result Q1=92[l/s], Q2=15[l/s], Q3=55[l/s] and Q4=23[l/s]. The Excel sheet is available on this link.