User:Egm6322.s09.Three.ge/MyIndividualProject

 See my comments below. Egm6322.s09 14:29, 1 May 2009 (UTC)

=Introduction=

Freshwater shortage is predicted by some to be one of the great problems of the 21st century. The desalination of salt water is one method of overcoming this problem, but currently consumes significant amounts of fossil fuels and as such is an unsustainable solution. The use of renewable energy to drive the desalination process is a sustainable method to overcome the problem of freshwater shortage. The use of solar energy as driving input in distillation offers promise since many water stressed areas receive significant solar energy.

Project Description and Goals
The aim of this project is to improve solar diffusion-driven stills by increasing the amount of distillate produced per unit area of still. The way this will be done is by optimizing the heat transfer in the still so that more heat is transferred via latent heat of evaporation. Furthermore, the recycling of heat released during condensation will be achieved by using multiple “effects”. A lab scale single distillation cell will be constructed in order to test the mathematical model.

References, Literature Review


The previous research in this area that is most comparable to this project was done primarily by Tanaka, et al. . In their work, they increased the amount of distillate produced by recycling the latent heat contained in the water vapor. The apparatus consisted of multiple distillation cells placed so that the latent heat of condensation was transferred to the saline water on the other side. An individual cell consisted of a plate receiving a heat flux, a wick on the other side of the plate, the vapor space, and finally the condenser plate which rejected heat. The heat on the evaporator plate drives water vapor off of the wick and across the vapor space to the condenser. At the condenser plate, the vapor condenses and heat is rejected through the plate to the evaporator on the other side. By reusing the latent heat of condensation to evaporate more water, the amount of freshwater distillate that can be produced in increased.

 It would be helpful to have a photo of a real multicell distillation apparatus with each components in a cell labeled. Egm6322.s09 14:32, 1 May 2009 (UTC)

=Modeling with PDEs=

The aim of this project is to increase the distillate production. To do this, the heat transferred by evaporation and condensation should be maximized, which may be accomplished by increasing the heat transferred through each of the walls. The heat conduction through a wall is governed by the heat equation.

The heat equation for Cartesian coordinates given by Özişik is:

$$\frac{\partial}{\partial x}\left ( k\frac{\partial T}{\partial x} \right )+\frac{\partial}{\partial y}\left ( k\frac{\partial T}{\partial y} \right )+\frac{\partial}{\partial z}\left ( k\frac{\partial T}{\partial z} \right )+ g= \rho C_{p} \frac{\partial T}{\partial t}$$

Where k is the thermal conductivity of the material, g is the energy generation within the material, Cp is the constant pressure specific heat of the material, and ρ is the material density.

For a plane wall that is thin a few key assumption can be made to simplify the heat equation.

Assumptions:


 * No heat generated

$$g=0$$


 * Negligible temperature gradients in y and z directions

$$\frac{\partial T}{\partial y}=\frac{\partial T}{\partial z}=0$$


 * Constant thermal conductivity, k

$$\frac{\partial }{\partial x}\left (k\frac{\partial T}{\partial x} \right )= k\frac{\partial^2 T}{\partial x^2}$$

The heat equation reduces to:

$$ k\frac{\partial^2 T}{\partial x^2}= \rho C_{p} \frac{\partial T}{\partial t} $$

Divide by k $$ \frac{\partial^2 T}{\partial x^2}= \frac {\rho C_{p}}{k} \frac{\partial T}{\partial t}$$

Note $$\alpha=\frac{k}{\rho C_{p}}$$

And the equation finally reduces to:

$$\frac{\partial^2 T}{\partial x^2}= \frac {1}{\alpha} \frac{\partial T}{\partial t} $$

Description of Boundary Conditions


The boundary conditions for this problem are the prescribed heat flux on one side and convective boundary on the other side (see illustration).

$$ \frac {\partial T}{\partial x}_{x=0}=q''$$

$$-k\frac {\partial T}{\partial x}_{x=L}=h(T-T_{amb})$$

$$T(t=0)=T_{0}$$

The final boundary condition assumes that the initial condition of the plate is a constant temperature, and that that temperature is the ambient temperature.

It should be noted that the boundary conditions are highly variable, depending on the solar heat flux and the convection coefficient of the saline water.

PDE Classification
The Heat equation PDE must first be put into canonical form by moving the time dependent term to the left hand side of the equation, resulting in:

$$\frac{\partial^2 T}{\partial x^2}-\frac {1}{\alpha} \frac{\partial T}{\partial t}=0$$

This form can then be classified according to the coefficients of the PDE's (see PDE classification).

It becomes clear that the coefficients of the heat equation are: $$a=1,b=0,c=0$$

Thus, $$ac-b^2=0$$ and the PDE is classified as parabolic.

This conclusion is further reinforced by comparison with the canonical form of a parabolic PDE.

$$\begin{matrix} \frac{\partial^2 T}{\partial x^2}-\frac {1}{\alpha} \frac{\partial T}{\partial t}=0\\ \xi^2-\eta = 0 \end{matrix}$$

Method of solution
The equation may be solved analytically by the method of separation of variables, but requires some reconfiguring. In order to solve the PDE the boundary conditions need to be shifted in order to make the convective boundary homogeneous, that is to say

$$-k\frac {\partial T}{\partial x}_{x=L}=h(T-T_{amb})\Rightarrow -k\frac {\partial T}{\partial x}_{x=L}=h\theta_{x=L}$$

Therefore:

$$\begin{matrix} \textrm{Let}\qquad \theta=T-T_{amb}\\ \frac{\partial^2 \theta}{\partial x^2}-\frac {1}{\alpha} \frac{\partial \theta}{\partial t}=0 \end{matrix}$$

And the PDE has the following boundary conditions:

$$\frac {\partial \theta}{\partial x}_{x=0}=q''$$

$$-k\frac {\partial \theta}{\partial x}_{x=L}=h\theta_{x=L}$$

$$\theta(t=0)=\theta_{0}$$

Since the PDE is linear, the method of superposition may be used to solve the equation. The PDE may be be described as the sum of a solution that does not depend on the time (a steady state solution, θs) and a solution whose boundary conditions are homogeneous (the homogeneous solution θh).

$$\theta(x,t)=\theta_{h}(x,t)+\theta_{ss}(x)$$

Once this has been done, the PDE may be solved using the method of Separation of Variables.

Solution Using Superposition and Separation of Variables

$$\frac{\partial^2 \theta_{ss}}{\partial x^2}=0$$
 * Steady State

Boundary Conditions

$$-k\frac{\partial^2 \theta_{ss}}{\partial x^2}=q''; \quad -k\frac{\partial \theta_{ss}}{\partial x}|_{x=L}=h\theta_{ss}|_{x=L}$$

Integrate:

$$\begin{matrix} \frac{\partial^2 \theta_{ss}}{\partial x^2}=0 \Rightarrow \int \partial(\frac{\partial \theta_{ss}}{\partial x})=\int0 \partial x \\ \therefore \\ \frac{\partial \theta_{ss}}{\partial x}=constant=\frac{-q''}{k} \end{matrix}$$

Integrate again and solve for the constants:

$$\begin{matrix} \theta_{ss}=\frac{-q''}{k}x+C_{0}\\ C_{0}=q''(hL-k-x)\\ \therefore \\ \theta_{ss}(x)=q''(hL-k-\frac{x}{k}) \end{matrix}$$


 * Homogeneous

$$\frac{\partial^2 \theta_{h}}{\partial x^2}=\frac {1}{\alpha} \frac{\partial \theta_{h}}{\partial t} $$

Boundary Conditions

$$\frac{\partial \theta_{h}}{\partial x}|_{x=0}=0; \quad -k\frac{\partial \theta_{h}}{\partial x}|_{x=L}=h\theta_{ss}|_{x=L} $$

And,

$$\theta_{h}(t=0)=\theta_{0}-\theta_{ss}(x)$$

=Results=

The results of the PDE are highly dependent on the boundary conditions. It should be noted that the heat flux, q", is generally a function of time and that the heat transfer coefficient, h, and ambient temperature, Tamb, may also vary in response to the heat flux.

With this simplified equation, and choosing constants that are representative of what may be seen in the experiment.

=Conclusion=

As was mentioned in the results, the mathematical model used in this report was a simplified one. Solar heat flux depends on time, and the convection coefficient depends on what is occurring it the rest of the still.

Furthermore, the focus of the research is not on monitoring the tempurature profile through the evaporator plate, but to maximize the production of freshwater distillate. In order to accomplish this, a mathematical model (simplified) that illustrates the coupling between the evaporator and condensor will be developed, and it will be compared to experimental data.

=References=

=Signed= Egm6322.s09.Three.ge 20:57, 30 April 2009 (UTC)