text:
I need help with this induction proof

1. Dec 11, 2012

snapgilmore

1. The problem statement, all variables and given/known data

Prove that 2^n > n^3 for every integer n >= 10

2. Relevant equations

a) Prove for 10,
b) assume for k
c) inductive step for k+1

3. The attempt at a solution

a)
2^10 = 1024
10^3 = 1000
1024>1000

b)Assume
2^k > k^3

c) 2^(k+1) > (k+1)^3
(2)(2^k) > (k+1)^3
(2^k) > ((k+1)^3)/2

and from here I'm having trouble solving for k >= 10.
Any help would be appreciated, thanks.

2. Dec 12, 2012

Staff: Mentor

The inequality above is what you need to show. You can't just assert it.
Your work should look like this:
2k+1 = 2 * 2k > 2 * k3 ...

You then need to show that 2k3 > (k + 1)3, at least for k ≥ 10.

It suffices to show that 2k3/(k + 1)3 > 1, which can be done using calculus.

Last edited: Dec 12, 2012
3. Dec 12, 2012

snapgilmore

Can you elaborate on how its enough to show (2k^3)/(k+1)^3 >1 ? I thought you needed to end up at k >=10?

4. Dec 12, 2012

SammyS

Staff Emeritus
You showed that the inequality holds for n = 10.

Then you show that if the inequality holds for n = k, where k is any number greater than or equal to 10, then the inequality holds for n = k+1 .

By doing this, you show that if it's true for n=10, then it's true for n=11 .
If it's true for n=11, then it's true for n=12 .

If it's true for n=12, then it's true for n=13 .

If it's true for n=13, then it's true for n=14 .

If it's true for n=14, then it's true for n=15 .

...
So assume that 2k > k3, for some integer k ≥ 10 .

From that show that 2k+1 > (k+1)3 .

5. Dec 12, 2012

Staff: Mentor

(2k3)/(k+1)3 >1 is equivalent to 2k3 > (k + 1)3.


text:
# Differential Equation with unit step function and Laplace transforms

1. Apr 15, 2013

### Mangoes

EDIT:

Nevermind I see what I did wrong near the end.

1. The problem statement, all variables and given/known data

$$x'' + 4x = f(t)$$

Where f(t) is 1 if t is between 0 and π, 0 if t > π. Initial conditions are x(0) = x'(0) = 0.

2. Relevant equations

Transform of a derivative:
$$L(f^{(n)}(t)) = s^nF(s) - s^{n-1}f(0) -...-f^{n-1}(0)$$

Transform of unit step function:
$$L(u(t-a)) = \frac{e^{-as}}{s}$$

3. The attempt at a solution

Letting u(t) be the step function, I wrote:

$$f(t) = 1 - u(t - π)$$

So I rewrote the equation as

$$x'' + 4x = 1 - u(t - π)$$

To take the Laplace transform of the left side, I used the transform of a derivative. Since the initial conditions are 0, the transform simplifies quite a bit. For the right side of the equation, I had to use the transform of the unit step function and the transform of 1.

$$s^2X(s) + 4X(s) = \frac{1}{s} - \frac{e^{-πs}}{s}$$

Factored and moved around a bit to get:

$$X(s)(s^2 + 4) = \frac{1}{s}(1 - e^{-πs})$$

$$X(s) = \frac{1 - e^{-πs}}{(s)(s^2 + 4)}$$

$$X(s) = \frac{1}{s(s^2 + 4)} - \frac{e^{-πs}}{s(s^2 + 4)}$$

At this point I had a choice to do either a convolution or use partial fractions. Since I hate partial fractions, I went for a convolution, but it didn't work out and I had to end up using partial fractions.

Making up a new name:

$$G(s) = \frac{1}{s(s^2 + 4)}$$

I used partial fractions and got this:

$$G(s) = \frac{1}{4s} - \frac{s}{4(s^2 + 4)}$$

With the above I can easily do the inverse Laplace transform of the first term in X(s). But for the second term, I wanted to use the fact that

$$u(t-a)f(t-a) = e^{-as}F(s)$$

where a = π and F(s) would be the same as G(s). The inverse Laplace transform of G(s) is

$$4 - \frac{1}{4}cos(2t)$$

And by just following the fact I just mentioned above, I would think that the Laplace transform of the second term in X(s) would be:

$$u(t - π)(4 - \frac{1}{4}cos(2(t - π))$$

cos(2t - 2pi) simplifies to cos(2t) due to periodicity of cosine.

So finally my final answer would be

$$f(t) = 4 - \frac{1}{4}cos(2t) - u(t - π)(4 - \frac{1}{4}cos(2t))$$

When I checked the answers, my answer was a little off. Normally, this really wouldn't bother me too much because there's a very good chance I made a small error somewhere, but I'm not too used to doing questions like these and I was wondering if anyone could find a really obvious glaring error the procedure I did. I have an exam tomorrow and it would be nice to make sure.

Last edited: Apr 15, 2013

format: plain


text:
# [texhax] TeX Queries (1)

Paul Isambert zappathustra at free.fr
Mon Jul 9 19:08:56 CEST 2012

Le 09/07/2012 17:19, Paul Stanley a écrit :
> Hi folks
> A few Tex queries from the TeXBook. I'm looking at the TeX source,
> http://net.ytu.edu.cn/share/%D7%CA%C1%CF/texbook.pdf :

> chapter 1, page 1, paragraph 1:
> English words like technology' stem from a Greek root <elipsis> which
> is an uppercase form of $\tau\epsilon\chi$.^^{TeX (actually \TeX),
> meaning of} ^^|\tau|^^|\epsilon|^^|\chi|
>
> Does ^^{ ... }' signify a margin note?  I'm calling it a margine note
> because it appears separate from the main text on the right edge of
> the page.

The whole situation with ^{...}' deals with the index; it is explained
lines 23907 and following of the sources. (That the expressions are in
the margin is only for proofreading.)

> chapter 2, page 3, paragraph 2:
> In the first place, there are two kinds of <elipsis> that shows up as
> something like {\tt\char'22}, and an apostrophe or right-quote that
> looks like {\tt\char'15} or {\tt\char'23}.
>
> \char' according to some web sources maps numeric values to their
> corresponding unicode characters.  in the above extract the macro is
> separated from the numeric value by an apostrophe. I've seen both a
> grave accent () and a double quote mark (") used in examples on the
> web. Do the symbols declare different things about the number that
> follows them?

The syntax is: \char<number>; and a number can be octal (prefixed with a
single quote) or hexadecimal (double quote), and of course decimal
(unprefixed). For the grave I can't see any usage but denoting a number
via a character, i.e. \a (meaning 97, with the backslash optional in
this case).

By the way, \char<number> simply typesets the character at position
<number> in the font and has nothing to do with Unicode (especially as
legacy TeX knows nothing of Unicode).

> finally, what's the difference between \eject' and \vfil\eject'?  I
> understand \ject' forces a page break.  A \vfil' prefix I hear
> improves/mitigates the visual effect of \ject' in certain circs.
> Unfortunately, the text doesn't explain exactly what form the
> improvmenet takes.  Any clues would be appreciated.

The \vfil means the page will be flushed upwards, unless equivalent or
stronger \vfil's are used elswhere. For instance:

paragraph1

paragraph2

\eject

will create a page with paragraph1 at the top and paragraph2 at the
bottom because the only adjustment possible to fill the page is to
stretch the interparagraph glue; \vfil avoids that.

Best,
Paul

`


text:
$$\renewcommand\AA{\unicode{x212B}}$$

# IqtFitMultiple v1¶

## Summary¶

Fits an *_iqt file generated by I(Q,t).

## Properties¶

Name

Direction

Type

Default

Description

InputWorkspace

Input

MatrixWorkspace

Mandatory

The _iqt.nxs InputWorkspace used by the algorithm

Function

InOut

Function

Mandatory

The function to use in fitting

FitType

Input

string

The type of fit being carried out

StartX

Input

number

0

The first value for X

EndX

Input

number

0.2

The last value for X

SpecMin

Input

number

0

Minimum spectra in the workspace to fit

SpecMax

Input

number

1

Maximum spectra in the workspace to fit

Minimizer

Input

string

Levenberg-Marquardt

The minimizer to use in fitting

MaxIterations

Input

number

500

The Maximum number of iterations for the fit

ConstrainIntensities

Input

boolean

False

If the Intensities should be constrained during the fit

ExtractMembers

Input

boolean

False

If true, then each member of the fit will be extracted, into their own workspace. These workspaces will have a histogram for each spectrum (Q-value) and will be grouped.

OutputResultWorkspace

Output

MatrixWorkspace

Mandatory

The output workspace containing the results of the fit data

OutputParameterWorkspace

Output

TableWorkspace

Mandatory

The output workspace containing the parameters for each fit

OutputWorkspaceGroup

Output

WorkspaceGroup

Mandatory

The OutputWorkspace group Data, Calc and Diff, values for the fit of each spectra

## Description¶

Fits an *_iqt file generated by Iqt using one of the specified functions. The functions available are either one or two exponentials ($$Intensity \times exp[-(x/\tau)]$$), a stretched exponential ($$Intensity \times exp[-(x/\tau)]\beta$$) or a combination of both an exponential and stretched exponential.

This routine was originally part of the MODES package.

## Usage¶

Example - Running IqtFitMultiple on an reduced workspace.

#Load in iqt data


text:
1. ## Ratio of Circle Radii that Share tangent lines

I don't know how to explain the arrangement, so I've attached an image. Forgive my notation, I'm not used to writing math problems.

The points that are black are known. The points that are red are unknown. $P_0T_{11}$ and $P_0T_{12}$ are tangent to BOTH circles. $P_1P_2$ is tangent only to circle 2 (the larger circle).

Conceptually, I think this is easy. The center of circle 2 ( $C_2$) lies on $P_0P_2$, and the radius is such that it is tangent to both $C_1$ and $C_2$. Without any other conditions, there would be an infinite number of solutions. However, there is an additional line that bounds circle 2. So, unless I am mistaken, there is now only one solution.

However, I don't know what I'm doing wrong. I've tried using similar triangles, but because neither $C_2$ nor the radius for circle 2 is known, I haven't been able to determine the ratio of the two triangles. In fact, everything I've tried seems to require a system of equations.

So I tried the following, where $d(\textbf{x}_0,\textbf{x}_1)$ is the distance formula between points $\textbf{x}_0$ and $\textbf{x}_1$:
$d(P_0,T_2_1)=d(P_0,T_2_2)$
$d(P_1,T_2_2)=d(P_1,T_3)$
$d(P_0,T_2_2)+d(T_2_2,P_1)=d(P_0,P_1)$

However, when I write the distance formulas for three dimensions, I end up with nine unknowns; three for each of the following points, $T_2_1$, $T_2_2$, and $T_3$.

So, I tried an additional three equations based on the distance of the center of circle 2 to each of the bounding lines using the following formula:
$d=\dfrac{\mid(\textbf{x}_0-\textbf{x}_1)\times(\textbf{x}_0-\textbf{x}_2)\mid}{\mid\textbf{x}_2-\textbf{x}_1\mid}$

I needed three more equations, so I tried parameterizing the equation for $P_0P_1$, yielding:
$x_{T_{22}}=x_{P_0}+t(x_{P_1}-x_{P_0})$
$y_{T_{22}}=y_{P_0}+t(y_{P_1}-y_{P_0})$
$z_{T_{22}}=z_{P_0}+t(z_{P_1}-z_{P_0})$

where $t=\dfrac{d(P_0,T_{22})}{d(P_0,P_1)}$

However, this led to a binomial equation in x, y and z. So, instead I tried using the equation for each of the three bounding lines, since the respective tangent point is a solution to the corresponding line equation. Unfortunately, I'm not sure if I'm not solving the system of equations correctly, or if my equations are not independent from each other.

Again, I feel like there should be an elegant solution to this. Perhaps something similar to finding the tangent points of a circle inscribed inside a triangle.

Any help would be greatly appreciated.

2. Hi arcanine, welcome to MHF.

P1 and P2 are known points.

Find the equation of the tangent P1P2.

Write it in the form y = mx + c.

Condition for line to be a tangent to a circle C2 is

$c^2 = r_{2}^2(1 + m^2)$

find r2. T11 is the radius of C1. Now find the ratio of r1 and r2.

3. ## I'm not sure I understand your post

Originally Posted by sa-ri-ga-ma
Find the equation of the tangent P1P2.

Write it in the form y = mx + c.
In this case, $P_1$ and $P_2$ have the same y-coordinates, so,
$y=c=y_{P_1}$

Condition for line to be a tangent to a circle C2 is

$c^2 = r_{2}^2(1 + m^2)$

find r2. T11 is the radius of C1. Now find the ratio of r1 and r2.
When using the above equation, I get a result that is not the radius of circle 2 (See Attached Image). Also note that Circle 3 does not pass through $C_1$, and if it did, it would only be a coincidence.

4. In the problem you have mensioned that P1 and P2 are known points. (The points that are black are known) Is it so?

Then how can be their y-coordinates are the same? In the picture they are not the same.

5. Originally Posted by sa-ri-ga-ma
In the problem you have mensioned that P1 and P2 are known points. (The points that are black are known) Is it so?
Yes, it is so. Perhaps my assumptions are not standard, but I was assuming that the y-axis was a vertical line on the screen, and the x-axis was a horizontal line on the screen, such that they are perpendicular to eachother. Under that assumption, the y-coords for $P_1$ and $P_2$ are the same.

However, I'm not too concerned with their y-coordinates. I'm more interested in how you derived the equation for a line tangent to a circle, because it is not giving the correct radius of the circle, as mentioned in post #3. An important note is that the origin is at an arbitrary location.

Without actually writing things out, it appears that the equation you provided for a line tangent to a circle assumes that the circle's center is at the origin. If this were the case, I wouldn't have any problems, because $C_2$ would become a known point.

6. Assuming $P_1P_2$ is horizontal, then $C_2T_3$ is vertical, and the equation for the line is $x = a$ for some value of $a$. Now the line $T_{22}C_2$ is perpendicular to $P_0P_1$ and therefore you know its slope $m_2$.

So since $P_0P_2$, $C_2T_3$ and $T_{22}C_2$ all intersect at the same point, you have:

$T_{22}C_2: y = m_2x + b_2$
$P_0P_2: y = m_1x + b_1$
$C_2T_3: x = a$
$P_1P_2: y = b$
$P_0P_1: y = m_3x + b_3$

Therefore the point of intersection can be written:
$(a, m_1a + b_1)$

where $a$ and $b_2$ are the only unknowns.

Now, $m_1a + b_1 = m_2a + b_2$, which yields
$b_2 = (m_1 - m_2)a + b_1$

Continuing, we find the point of intersection of $P_0P_1$ and $T_{22}C_2$ in terms of a:

$m_2x + b_2 = m_3x + b_3$
$m_2x + (m_1 - m_2)a + b_1 = m_3x + b_3$
$(m_2 - m_3)x = b_3 - (m_1 - m_2)a$
$x_0 = \frac{b_3 - (m_1 - m_2)a}{m_2 - m_3}$
$y_0 = \frac{m_3(b_3 - (m_1 - m_2)a)}{m_2 - m_3} + b_3$

Now the distances $T_{22}C_2$ and $T_3C_2$ must be equal, therefore $\sqrt{(a - x_0)^2 + (m_1a + b_1 - y_0)^2} = m_1a + b_1 - b$. Since everything is in terms of a or known constants, solve the equation for a and then you will know the location of every unknown point.

7. As you have guessed, I have assumed that Po is the origin. When you say that some points are known, you must mention the poisition of the origin.

8. Originally Posted by sa-ri-ga-ma
When you say that some points are known, you must mention the poisition of the origin.
I apologize, I'm new to describing math problems.

I've looked at icemanfan's solution, and I believe it will work. I haven't crunched through the math to be sure, but at first glance it looks good.

Thanks everyone for the help.


text:
# Algebra 1

Sub Topics

Algebra 1 takes through the basics of indices and exponents, functions, relations, theory of equations, algebraic expressions, quadratic equations and factorization methods.

## Algebra 1 Formulas

Algebra basics formula helps in speeding up the calculation part. Some of the algebra formula include:

(a+b)= a+ b2 + 2ab

(a-b)= a+ b- 2ab

(a+b)3 = a3 +b+ 3ab (a+b)

(a-b)= a- b- 3ab (a-b)

## Algebra 1 Topics

Choose your Algebra 1 topics below:

Chapter 1: Algebra basics

Chapter 2:Linear Functions

Chapter 3: Inequalities

Chapter 4: Systems of Equations

Chapter 5: Polynomials

Chapter 6: Exponents

Chapter 8: Lines

Chapter 10: Complex Numbers

Chapter 11: Algebraic Fractions (Rational Functions)

## Algebra 1 Problems

Below you could see examples

### Solved Examples

Question 1: solve for x,

x - $\frac{1}{2}$ = 2
Solution:
Given: x - $\frac{1}{2}$ = 2

Take $\frac{1}{2}$ on both sides

x = 2 + $\frac{1}{2}$

x = $\frac{5}{2}$

Question 2: Factorize, x2 + 5x + 6 = 0
Solution:
Given: x2 + 5x + 6 = 0

x2 + 3x + 2x +6 = 0

x(x + 3) + 2( x + 3) = 0

(x + 3) (x + 2) = 0

x = -3

x = -2

## Algebra 1 Homework Help

Besides tutoring TutorNext also offers regular homework help and assignment help where a student can upload assignments on our website and get a detailed, completed assignment with step by step explanations within 48 hours. In case you have any doubts or need further help you can just connect with a tutor who can give you all the support with math homework help like Algebre homework help, algebra 1 homework help and Algebra 2 homework help. Avail Algebra 2 help from our expert online tutors.

Algebra instant help with TutorNext's Algebra 1 tutoring covers all topics for K-12 and beyond and much more. Our tutors are experts in their subject and have many years of experience tutoring students across grades, they can handle any doubts and queries on math help algebra 1. They also undergo an training program to ensure that you get the best tutoring possible. Our Algebra 1 tutors follow the same curriculum as followed in your school there by ensuring that there is continuity with what you are learning in class. In fact our online tutoring is so personalized that it is like having an online Algebra 1 Helper besides you.

## Algebra 1 Practice

Some questions for practice:

### Practice Problems

Question 1: Solve for x,

3x - 9 = 18.
Question 2: Factorize x2 - 5x + 6 = 0
Question 3: Solve the system of equations:

2x - 5y = 4

3x - 2y = -1.


text:
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New Member

## inline power for 7960

Hi

Does anyone know if the distance limitation for a 7960 with inline power is the same as the normal UTP distance limitation for ethernet?

Thanks,

Larry

1 ACCEPTED SOLUTION

Accepted Solutions
Bronze

## Re: inline power for 7960

yes. i have a couple of phones about 350 feet away (a bit longer than recommended) but they work fine.

Geoff

2 REPLIES
Bronze

## Re: inline power for 7960

yes. i have a couple of phones about 350 feet away (a bit longer than recommended) but they work fine.

Geoff

New Member

thanks Geoff

98
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0


text:
# Summarizing fixed effects variables in Stata

Using a data set of 209 countries, I have run a fixed effects regression for non-OECD countries.

When I summarize my variables, it gives me all of the variables in data set, but I only need a summary of those variables that have been used in my fixed effects regression. For instance, when I run fixed effects, the number of groups is 138. So, my total observations for a variable should be 138 multiplied by 26 years (1980-2006), which is only 3588 observations.

How can I ask Stata to generate this?

-

Stata leaves the quasi-variable e(sample) after any estimation command. You can use it as summarize x1 x2 x3 if e(sample).
In general, you should develop a habit of looking into the guts of Stata with ereturn list after estimation. You may be able to find a lot of useful additional information.
-estat summarize- will give summaries of the sample used in the most recent estimation command. See -help postestimation-, or the postestimation help for the specific command you are using.


text:
# 2 simple trig deriv's

• February 23rd 2010, 04:55 PM
maybnxtseasn
2 simple trig deriv's
sinxy²

i got 2cosxy

--------------

sin²xy

i got 2sinxy*(cosxy+sinx)
• February 23rd 2010, 05:04 PM
Prove It
Quote:

Originally Posted by maybnxtseasn
sinxy²

i got 2cosxy

--------------

sin²xy

i got 2sinxy*(cosxy+sinx)

These are very hard to read. Please either use brackets or LaTeX.

Is your first function $\sin{(xy^2)}$ or $(\sin{x})y^2$?

Is your second function $\sin^2{(xy)}$ or $(\sin^2{x})y$?
• February 23rd 2010, 05:22 PM
maybnxtseasn
Quote:

Originally Posted by Prove It
These are very hard to read. Please either use brackets or LaTeX.

Is your first function $\sin{(xy^2)}$ or $(\sin{x})y^2$?

Is your second function $\sin^2{(xy)}$ or $(\sin^2{x})y$?

$\sin{(xy^2)}$
$\sin^2{(xy)}$


text:
# How do I find if a number is a multiple of 8?

Apr 4, 2016

Please see the process and reason below.

#### Explanation:

Just check the last three digits of the number.

If they are divisible by $8$, the entire number is divisible by $8$.

For example, last three digits of $53927592$ are $592$ and they are divisible by $8$ as $592 = 74 \times 8$.

Hence, $53927592$ is divisible by $8$ and $53927592 = 6740949 \times 8$.

The reason is that as the number can be expressed as
$Y Y Y Y Y Z Z Z = Y Y Y Y Y \times 1000 + Z Z Z$

= $Y Y Y Y Y \times 125 \times 8 + Z Z Z$.

As $Y Y Y Y Y \times 125 \times 8$ is divisible by $8$, if $Z Z Z$ is divisible by $8$, $Y Y Y Y Y Z Z Z$ is divisible by $8$.


text:
## C Specification

The depth values of all fragments generated by the rasterization of a polygon can be offset by a single value that is computed for that polygon. This behavior is controlled by the depthBiasEnable, depthBiasConstantFactor, depthBiasClamp, and depthBiasSlopeFactor members of VkPipelineRasterizationStateCreateInfo, or by the corresponding parameters to the vkCmdSetDepthBias command if depth bias state is dynamic.

void vkCmdSetDepthBias(
VkCommandBuffer                             commandBuffer,
float                                       depthBiasConstantFactor,
float                                       depthBiasClamp,
float                                       depthBiasSlopeFactor);

## Parameters

• commandBuffer is the command buffer into which the command will be recorded.

• depthBiasConstantFactor is a scalar factor controlling the constant depth value added to each fragment.

• depthBiasClamp is the maximum (or minimum) depth bias of a fragment.

• depthBiasSlopeFactor is a scalar factor applied to a fragment’s slope in depth bias calculations.

## Description

If depthBiasEnable is VK_FALSE, no depth bias is applied and the fragment’s depth values are unchanged.

depthBiasSlopeFactor scales the maximum depth slope of the polygon, and depthBiasConstantFactor scales an implementation-dependent constant that relates to the usable resolution of the depth buffer. The resulting values are summed to produce the depth bias value which is then clamped to a minimum or maximum value specified by depthBiasClamp. depthBiasSlopeFactor, depthBiasConstantFactor, and depthBiasClamp can each be positive, negative, or zero.

The maximum depth slope m of a triangle is

$m = \sqrt{ \left({{\partial z_f} \over {\partial x_f}}\right)^2 + \left({{\partial z_f} \over {\partial y_f}}\right)^2}$

where (xf, yf, zf) is a point on the triangle. m may be approximated as

$m = \max\left( \left| { {\partial z_f} \over {\partial x_f} } \right|, \left| { {\partial z_f} \over {\partial y_f} } \right| \right).$

The minimum resolvable difference r is an implementation-dependent parameter that depends on the depth buffer representation. It is the smallest difference in framebuffer coordinate z values that is guaranteed to remain distinct throughout polygon rasterization and in the depth buffer. All pairs of fragments generated by the rasterization of two polygons with otherwise identical vertices, but zf values that differ by r, will have distinct depth values.

For fixed-point depth buffer representations, r is constant throughout the range of the entire depth buffer. For floating-point depth buffers, there is no single minimum resolvable difference. In this case, the minimum resolvable difference for a given polygon is dependent on the maximum exponent, e, in the range of z values spanned by the primitive. If n is the number of bits in the floating-point mantissa, the minimum resolvable difference, r, for the given primitive is defined as

r = 2e-n

If a triangle is rasterized using the VK_POLYGON_MODE_FILL_RECTANGLE_NV polygon mode, then this minimum resolvable difference may not be resolvable for samples outside of the triangle, where the depth is extrapolated.

If no depth buffer is present, r is undefined.

The bias value o for a polygon is

$o = \begin{cases} m \times depthBiasSlopeFactor + r \times depthBiasConstantFactor & depthBiasClamp = 0\ or\ NaN \\ \min(m \times depthBiasSlopeFactor + r \times depthBiasConstantFactor, depthBiasClamp) & depthBiasClamp > 0 \\ \max(m \times depthBiasSlopeFactor + r \times depthBiasConstantFactor, depthBiasClamp) & depthBiasClamp < 0 \\ \end{cases}$

m is computed as described above. If the depth buffer uses a fixed-point representation, m is a function of depth values in the range [0,1], and o is applied to depth values in the same range.

For fixed-point depth buffers, fragment depth values are always limited to the range [0,1] by clamping after depth bias addition is performed. Unless the ../../html/vkspec.html#VK_EXT_depth_range_unrestricted extension is enabled, fragment depth values are clamped even when the depth buffer uses a floating-point representation.

Valid Usage
• The bound graphics pipeline must have been created with the VK_DYNAMIC_STATE_DEPTH_BIAS dynamic state enabled

• If the depth bias clamping feature is not enabled, depthBiasClamp must be 0.0

Valid Usage (Implicit)
• commandBuffer must be a valid VkCommandBuffer handle

• commandBuffer must be in the recording state

• The VkCommandPool that commandBuffer was allocated from must support graphics operations

Host Synchronization
• Host access to commandBuffer must be externally synchronized

• Host access to the VkCommandPool that commandBuffer was allocated from must be externally synchronized

Command Properties
Command Buffer Levels Render Pass Scope Supported Queue Types Pipeline Type

Primary
Secondary

Both

Graphics


text:
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text:
Open access peer-reviewed chapter

# Catalytic Hydroprocessing of Liquid Biomass for Biofuels Production

By Stella Bezergianni

Submitted: April 17th 2012Reviewed: August 23rd 2012Published: March 20th 2013

DOI: 10.5772/52649

## 1. Introduction

The depletion of world petroleum reserves and the increased concern on climate change has stimulated the recent interest in biofuels. The most common biofuels are based on energy crops and their products, i.e. vegetable oil for Fatty Acid Methyl Esters (FAME) biodiesel [1] and sugars/starch for bioethanol. However these first generation biofuels and associated production technologies face several considerations related to their economic and social implications regarding energy crops cultivation, by-products disposal, necessity for large investments to ensure competitiveness and the “food versus fuel” debate.

As a result, second generation biofuel technologies have been developed to overcome the limitations of first generation biofuels production [2]. The goal of second generation biofuel processes is to extend biofuel production capacity by incorporating residual biomass while increasing sustainability. This residual biomass consists of the non-food parts of food crops (such as stems, leaves and husks) as well as other non-food crops (such as switch grass, jatropha, miscanthus and cereals that bear little grain). Furthermore the residual biomass potential is further augmented by industrial and municipal organic waste such as skins and pulp from fruit pressing, waste cooking oil etc. One such technology is catalytic hydroprocessing, which is an alternative conversion technology of liquid biomass to biofuels that is lately raising a lot of interest in both the academic and industrial world and is the proposed subject of this chapter.

Catalytic hydroprocessingis a key process in petrochemical industry for over a century enabling heteroatom (sulfur, nitrogen, oxygen, metals) removal, saturation of olefins and aromatics, as well as isomerization and cracking [3]. Due to the numerous applications of catalytic hydroprocessing, there are several catalytic hydroprocessing units in a typical refinery including distillate hydrotreaters and hydrocrackers (see Figure 1). As a result several refinery streams are treated with hydrogen in order to improve final product quality including straight-run naphtha, diesel, gas-oils etc. The catalytic hydroprocessing technology is evolving through the new catalytic materials that are being developed. Even though hydroprocessing catalysts development is well established [4], the growing demand of petroleum products and their specifications, which are continuously becoming stricter, have created new horizons in the catalyst development in order to convert heavier and lower quality feedstocks [5]. Furthermore the expansion of the technology to bio-based feedstocks has also broadened the R&D spam of catalytic hydrotreatment.

Catalytic hydroprocessing of liquid biomass is a technology that offers great flexibility to the continuously increasing demands of the biofuels market, as it can convert a wide variety of liquid biomass including raw vegetable oils, waste cooking oils, animal fats as well as algal oils into biofuels with high conversion yields. In general this catalytic process technology allows the conversion of triglycerides and lipids into paraffins and iso-paraffins within the naphtha, kerosene and diesel ranges. The products of this technology have improved characteristics as compared to both their fossil counterparts and the conventional biofuels including high heating value and cetane number, increased oxidation stability, negligible acidity and increased saturation level. Besides the application of this catalytic technology for the production of high quality paraffinic fuels, catalytic hydroprocessing is also an effective technology for upgrading intermediate products of solid biomass conversion technologies such as pyrolysis oils and Fischer-Tropsch wax (Figure 2). The growing interest and investments of the petrochemical, automotive and aviation industries to the biomass catalytic hydroprocessing technology shows that this technology will play an important role in the biofuels field in the immediate future.

In the sections that follow, the basic technical characteristics of catalytic hydrotreatment are presented including a description of the process, reactions, operating parameters and feedstock characteristics. Furthermore key applications of catalytic hydroprocessing of liquid biomass are outlined based on different feedstocks including raw vegetable oils, waste cooking oils, pyrolysis oils, Fischer-Tropsch wax and algal oil, and some successful demonstration activities are also presented.

## 2. Technical characteristics of catalytichydrotreatment

The catalytic hydrotreatment of liquid biomass converts the contained triglycerides/lipids into hydrocarbons at high temperatures and pressures over catalytic material under excess hydrogen atmosphere. The catalytic hydrotreatment of liquid biomass process is quite similar to the typical process applied to petroleum streams, as shown in Figure 3. A typical catalytic hydrotreatment unit consists of four basic sections: a) feed preparation, b) reaction, c) product separation and d) fractionation.

In the feed preparation section the liquid biomass feedstock is mixed with the high pressure hydrogen (mainly from gas recycle with some additional fresh make-up hydrogen) and is preheated before it enters the reactor section. The reactor section consists normally of two hydrotreating reactors, a first guard mild hydrotreating reactor and a second one where the main hydrotreating reactions take place. Each reactor contains two or more catalytic beds in order to maintain constant temperature profile throughout the reactor length. Within the reactor section all associated reactions take place, which will be presented in more detail at a later paragraph.

The reactor product then enters the separator section where, after it is cooled down, it enters the high pressure separator (HPS) flash drum in which the largest portion of the gas and liquid product molecules are separated. The gas product of the HPS includes the excess hydrogen that has not reacted within the reactor section as well as the side products of the reactions including CO, CO2, H2S, NH3 and H2O. The liquid product of the HPS is lead to a second flash drum, the low pressure separator (LPS), for removing any residual gas contained in the liquid product, and subsequently is fed to a fractionator section. The fractionator section provides the final product separation into the different boiling point fractions that yield the desired products including off-gas, naphtha, kerosene and diesel. The heaviest molecules return from the bottom of the fractionator into the reactor section as a liquid recycle stream.

In order to improve the overall efficiency, a liquid recycle stream is also incorporated, which in essence consists of the heavy molecules that were not converted. The gas product from the HPS and LPS, after being treated to remove the excess NH3, H2S, CO and CO2, is compressed and fed back to the reactor section as a gas recycle stream in order to maintain a high pressure hydrogen atmosphere within the reactor section.

### 2.1. Reaction mechanisms

Several types of reactions take place during catalytic hydrotreatment of liquid biomass, based on the type of biomass processed, operating conditions and catalyst employed. The types of reactions that liquid biomass undergoes during catalytic hydroprocessing include: a) cracking, b) saturation, c) heteroatom removal and d) isomerization, which are described in more detail in the following section.

#### 2.1.1. Cracking

As the molecules included in the various types of liquid biomass can be relatively large and complicated, cracking reactions are desired to convert them into molecules of the size and boiling point range of conventional fuels, mainly gasoline, kerosene and diesel. A characteristic reaction that occurs during catalytic hydrotreating of oils / fats is the cracking of triglycerides into its consisting fatty acids (carboxylic acids) and propane as shown in Scheme 1 [5][6]. This reaction is critical as it converts the initial large triglycerides molecules of boiling point over 600°C into mid-distillate range molecules (naphtha, kerosene and diesel).

Other cracking reactions may take place however such as those described in Schemes 2 and 3, depending on the type of molecules present in the feedstock. For example Scheme 2 is a cracking reaction which may occur during catalytic hydrotreatment of pyrolysis oil which includes polyaromatic and aromatic compounds. Alternatively Scheme 3 may follow deoxygenation of carboxylic acids on the produced long chain paraffinic molecules, leading to smaller chain paraffins, during the upgrading of Fischer-Tropsch wax.

#### 2.1.2. Saturation

Saturation reactions are strongly associated with catalytic hydrotreating as the introduction of excess hydrogen allows the breakage of double C-C bonds and their conversion to single bonds, as shown in the following reactions. In particular the saturation of unsaturated carboxylic acids into saturated ones depicted in Scheme 4, is a key reaction occurring in lipid feedstocks. Furthermore other saturation reactions lead to the formation of naphthenes by converting unsaturated cyclic compounds and aromatic compounds as in Scheme 5 and 6, which are likely to occur during upgrading of pyrolysis oils.

#### Scheme 6.

As a result of this reaction the produced saturated molecules are less active and less prone to polymerization and oxidation reactions, mitigating the sediment formation and corrosion phenomena appearing in engines.

#### 2.1.3. Heteroatom removal

Heteroatoms are atoms other than carbon (C) and hydrogen (H) and are often encountered into bio- and fossil- based feedstocks. They include sulfur (S), nitrogen (N) and in the case of bio-based feedstocks oxygen (O). In particular oxygen removal is of outmost importance as the presence of oxygen reduces oxidation stability (due to carboxylic and carbonylic double bonds), increases acidity and corrosivity (due to the presence of water) and even reduces the heating value of the final biofuels. The main deoxygenation reactions that take place include deoxygenation, decarbonylation and decarboxylation presented in Schemes 7, 8 and 9 respectively [7]. The main products of deoxygenation reactions include n-paraffins, while H2O, CO2 and CO are also produced, but can be removed with the excess hydrogen within the flash drums of the product separation section. It should be noted however that these particular reactions give the paraffinic nature of the produced biofuels, and for this reason the hydrotreated products are often referred to as paraffinic fuels (e.g. paraffinic jet, paraffinic diesel etc)

#### Scheme 9.

The other heteroatoms, i.e. S and N are removed according to the classic heteroatom removal mechanisms of the fossil fuels in the form of gaseous H2S and NH3 respectively.

#### 2.1.4. Isomerization

The straight chain paraffinic molecules resulting from the aforementioned reactions, even though they offer increased cetane number, heating value and oxidation stability in the biofuels which contain them, they also degrade their cold flow properties. In order to improve the cold flow properties, isomerization reactions are also required, which normally take place during a second step/reactor as they require a different catalyst. Some examples of isomerization reactions are given in Schemes 10 and 11.

### 2.2. Hydroprocessing catalysts

Catalytic hydroprocessing of liquid biomass is a technology currently under developed and there is a lot of room for optimization. For example there are not many commercial catalysts specifically designed and developed for such applications, while conventional commercial catalysts, employed for catalytic hydroprocessing of refinery streams, are used instead. Common hydrotreating catalysts employed contain active metals on alumina substrate with increased surface area. The most known commercial catalysts employ Cobalt and Molybdenum (CoMo) or Nikel and Molybdenum (NiMo) in alumina substrate (Al2O3) as shown in Figure 4.

Hydrotreating catalysts are dual action catalytic material, triggering both hydrogenation and cracking/isomerization reactions. On one hand hydrogenation takes place on the active metals (Mo, Ni, Co, Pd, Pt) which catalyze the feedstock molecules rendering them more active when subject to cracking and heteroatom removal, while limiting coke formation on the catalyst. Furthermore hydrogenation supports cracking by forming an active olefinic intermediate molecule via dehydrogenation. On the other hand both cracking and isomerization reactions take place in acidic environment such as amorphous oxides (SiO2 – Al2O3) or crystalline zeolites (mainly z-zeolites) or mixtures of zeolites with amorphous oxides.

During the first contact of the feedstock molecules with the catalyst, a temperature increase is likely to develop due to the exothermic reactions that occur. However, during the continuous utilization of the catalyst and coke deposition, the catalyst activity eventually reduces from 1/3 to 1/2 of its initial one. The catalyst deactivation rate mainly depends on temperature and hydrogen partial pressure. Increased temperatures accelerate catalyst deactivation while high hydrogen partial pressure tends to mitigate catalyst deactivation rate. Most of the catalyst activity can be recovered by catalyst regeneration.

The selection of a suitable hydroprocessing catalyst is a critical step defining the hydroprocessing product yield and quality as well as the operating cycle time of the process in petroleum industry [5]. However the hydrotreating catalyst selection for biomass applications is particularly crucial and challenging for two reasons: a) catalyst activity varies significantly, as commercial catalysts are designed for different feedstocks, i.e. feedstocks with high sulfur concentration, heavy feedstocks (containing large molecules), feedstocks with high oxygen concentration etc, and b) there are currently no commercial hydroprocessing catalysts available for lipid feedstocks and other intermediate products of biomass conversion processes (e.g. pyrolysis biooil), and thus commercial hydrotreating catalysts need to be explored and evaluated as different catalyst have different yields (Figure 5) and different degradation rate [8]. Nevertheless, significant efforts have been directed towards developing special hydrotreating catalysts for converting/upgrading liquid biomass to biofuels [9-12].

### 2.3. Operating parameters

As it was mentioned earlier, the choice of catalyst and operating parameters affect the reactions that take place within the hydroprocessing reactor. The key operating parameters of hydroprocessing include the reactor temperature, hydrogen partial pressure, liquid hourly space velocity and hydrogen feed-rate.

#### 2.3.1. Temperature

Most catalytic hydrotreating and hydrocracking reactors operate between 290-450°C. The temperature range is selected according the type of catalyst and feedstock type to be processed. In the first stages of the catalyst life (after its loading in the reactor) the temperature is normally kept low as the catalyst activity is already very high. However as time progresses and the catalyst deactivates and cokes, the temperature is gradually increased to overcome the loss of catalyst activity and to maintain the desired product yield and quality.

#### 2.3.2. Hydrogen partial pressure

Hydrogen partial pressure affects significantly the hydrotreating reactions as well as the catalyst deactivation. The catalyst deactivation rate is inverse proportional to the hydrogen partial pressure and to hydrogen feed-rate. However high hydrogen partial pressures correspond to high operational costs, which rise even higher for high olefinic feedstocks that exhibit higher hydrogen consumption due to the saturation reactions. Therefore hydrogen partial pressure should be balanced with the catalyst activity and catalyst life expectancy in order to optimize the overall process.

#### 2.3.3. Liquid hourly space velocity

Liquid hourly space velocity (LHSV) is defined as the ratio of the liquid mass feed-rate (gr/h) over the catalyst mass (gr) and as a result is expressed in hr-1. In fact the inverse of LHSV is proportional to the residence time of the liquid feed in the reactor. In essence the higher the liquid hourly space velocity, the less time is available for the contact of the feed molecules of the reaction mixture with the catalyst, thus the less the conversion. However, maintaining large LHSV imposes a faster degradation of the catalyst therefore in industrial applications the LHSV is maintained in as high values as it is practically possible.

#### 2.3.4. Hydrogen feed-rate

The hydrogen feed-rate is another important parameter as it also defines hydrogen partial pressure depending on the hydrogen consumption of each application. It actually favours both heteroatom removal and saturation reaction rates. However, as hydrogen cost defines the overall unit operating cost, hydrogen feed-rate is normally optimized depending on the system requirements. Furthermore the use of renewable energy sources for hydrogen production is also envisioned as a potential cost improvement option.

## 3. Feed sand products

Even though liquid biomass is currently being exploited as a renewable feedstock for fuels production, its characteristics are far beyond suitable for its use as fuel. More specifically liquid biomass, just as other types of biomass, has a small H/C ratio and high oxygen content, lowering its heating value and increasing CO and CO2 emissions during its combustion. Moreover liquid biomass contains water, which can cause corrosion in the downstream processing units if it’s not completely removed, or even in the engine parts where its final products are utilized. In addition to the above, liquid biomass has an increased concentration in oxygenated compounds, mainly acids, aldehydes, ketones etc, which not only reduce the heating value, but also decrease the oxidation stability and increase the acidity of the produced biofuels. For all the aforementioned reasons it is imperative that liquid biomass should be upgraded and specifically that its H/C should be increased while the water and oxygen removed.

The effectiveness of catalytic hydroprocessing towards improving these problematic characteristics of liquid biomass is presented in Table 1, where the H/C ratio, the oxygen content and density before and after catalytic hydrotreatment of basic liquid biomass types are given. The H/C ratio exhibits a significant increase that exceeds 50% in all cases. This is due to the substitution of the heteroatoms by hydrogen atoms as well as in the saturation of double bonds that enriches the H/C analogy. The oxygen content (including the oxygen contained in the water) from over 15%wt can be decreased down to 5wppm. Actually the deep deoxygenation achieved via catalytic hydrotreatment is the most significant contribution of this biomass conversion technology, as it improves significantly the oxidation stability of the final biofuels. Furthermore significant improvement is also observed in the biomass density, which is never below 0.9 kg/l while after hydrotreatment it reduces to values less than 0.8 kg/l

 Liquid biomass(unprocessed) Hydrotreated liquid biomassand produced biofuels H/C ratio 0.08 – 0.1 0.13 – 0.18 Oxygen content (%wt) 15 - 40 10-4 – 3 Density (kg/l) 0.9 – 1.05 0.75 – 0.8

### Table 1.

Effect of catalytic hydrotreatment on the liquid biomass characteristics

Catalytic hydroprocessing has been proven as the most efficient technology for the upgrading of liquid biomass as it achieves to increase the H/C ratio and to remove oxygen and water. However the effectiveness of this technology is also shown in other parameters. For example the distillation curve of raw liquid biomass shows that over 90% of its molecules have boiling points exceeding 600°C and only 5% are within diesel range (220-360°C), while after catalytic hydrotreatment upgrading most of 90% of the product molecules are within diesel range [13].

In the following sections the basic types of liquid biomass and their corresponding products via catalytic hydrotreatment are presented.

### 3.1. Raw vegetable oils conversion to paraffinic biofuels

Vegetable oils are the main feedstock for the production of first generation biofuels, which can offer several CO2 benefits and limit the consumption of fossil fuels. Raw vegetable oils consist of fatty acid triglycerides, the consistency of which depends on their origin (i.e. plant type) as shown in Table 2. Their production, however, is competing for the cultivated areas that were originally dedicated for the production of food and feed crops. As a result the production and utilization of vegetable oils for biofuels production has instigated the “food vs. fuel” debate. For this reason traditional energy crops (soy, cotton, etc) with low oil yield per hectare are being substituted by new energy crops (eg. jatropha, palm, castor etc).

 C8:0 C10:0 C12:0 C14:0 C16:0 C16:1 C18:0 C18:1 C18:2 C18:3 C20:0/C22:0 C20:1/C22:1 EU Rapeseed oil 0.0 0.0 0.0 0.0 3.5 1.0 1.5 12.5 15.0 7.5 9.0 50.0 Soybean oil 0.0 0.0 0.0 0.3 8.2 0.5 4.5 25.0 49.0 5.0 7.5 0.0 Sunflower oil 0.0 0.0 0.0 0.0 6.0 0.0 4.2 18.8 69.3 0.3 1.4 0.0 Corn oil 0.0 0.0 0.0 1.0 9.0 1.5 2.5 40.0 45.0 0.0 0.0 1.0 non-EU Palm oil 0.0 0.0 0.0 3.5 39.5 0.0 3.5 47.0 6.5 0.0 0.0 0.0 Peanut oil 0.0 0.0 0.0 0.5 8.0 1.5 3.5 51.5 27.5 0.0 7.5 0.0 Canola oil 0.0 0.0 0.1 0.1 4.7 0.1 1.6 65.9 21.2 5.2 1.2 0.0 Castor oil 0.0 0.1 0.2 10.6 1.4 9.5 29.7 29.7 41.3 3.3 3.8 0.0

### Table 2.

Fatty acid composition of most common vegetable oils [14][15]

Catalytic hydrotreatment was explored for conversion of vegetable oils in the early 90’s. The investigation of the hydrogenolysis of various vegetable oils such as maracuja, buritimtucha and babassu oils over a Ni–Mo/γ-Al2O3 catalyst as well as the effect of temperature and pressure on its effectiveness was firstly investigated [16][17]. The reaction products included a gas product rich in the excess hydrogen, carbon monoxide, carbon dioxide and light hydrocarbons as well as a liquid organic product of paraffinic nature. In more detail these studies showed the conversion of triglycerides into carboxyl oxides and then to high quality hydrocarbons via decarboxylation and decarbonylation reactions. Rapeseed oil hydroprocessing was also studied in lab-scale reactor for temperatures 310° and 360°C and hydrogen pressures of 7 and 15 MPa using three different Ni–Mo/alumina catalysts [18]. These products contained mostly n-heptadecane and n-octadecane accompanied by low concentrations of other n-alkanes and i-alkanes [19].

### 3.2. Waste cooking oils conversion to paraffinic biofuels

Even though vegetable oils are the main feedstock for the production of first generation biofuels, soon their production has troubled the public opinion due to their abated sustainability and to their association with the food vs. fuel debate. As a result the technology has shifted towards the exploitation of both solid and liquid residual biomass. Waste Cooking Oils (WCOs) is a type of residual biomass resulting from frying with typical vegetable frying oils (e.g. soybean-oil, corn-oil, olive-oil, sesame-oil etc). WCOs have particular problems regarding their disposal. In particular grease may result in coating of pipelines within the residential sewage system and is one of the most common causes of clogs and sewage spills. Furthermore, in the cases that sewage leaks into the environment, WCOs can cause human and environmental health problems because of the pathogens contained. It has been estimated that by disposing 1 lit of WCO, over 1,000,000 of liters of water can be contaminated, which is estimated as the average demand of a single person for 14 years.

Catalytic hydroprocessing of WCO was studied as an alternative approach of producing 2nd generation biofuels [20-24]. Initially catalytic hydrocracking was investigated over commercial hydrocracking catalysts leading not only to biodiesel but also to lighter products such as biogasoline [20], employing a continuous-flow catalytic hydroprocessing pilot-plant with a fixed-bed reactor. During this study several parameters were considered including hydrocracking temperature (350-390°C) and liquid hourly space velocity or LHSV (0.5-2.5 hr-1) under high pressure (140 bar), revealing that the conversion is favoured by high reaction temperature and low LHSV. Lower and medium temperatures, however, were more suitable for biodiesel production while higher temperatures offered better selectivity for biogasoline production. Furthermore, heteroatom removal (S, N and particularly O) was increased while saturation of double bonds was decreased with increasing hydrocracking temperature, indicating the necessity of a pre-treatment step.

However catalytic hydrotreatment was later examined in more detail as a more promising technology particularly for paraffinic biodiesel production (Figure 1). The same team has studied the effect of temperature (330-398°C) on the product yields and heteroatom removal [21]. The study was conducted in the same pilot plant utilizing a commercial NiMo/Al2O3 hydrotreating catalyst over lower pressure (80 bar). According to this study, the hydrotreating temperature is the key operating parameter which defines the catalyst effectiveness and life. In fact lower temperatures (330°C) favour diesel production and selectivity. Sulfur and nitrogen removal were equally effective at all temperatures, while oxygen removal and saturation of double bonds were favoured by hydrotreating temperature. The same team also studied the effect of the other three operating parameters i.e. pressure, LHSV and H2/WCO ratio [22]. Moreover they also studied the hydrocarbon content of the products [23] qualitatively via two-dimensional chromatography and quantitatively via Gas Chromatography with Flame Ionization Detector (GC-FID), which indicated the presence of C15-C18 paraffins. Interestingly this study showed that as hydrotreating temperature increases, the content of normal paraffins decreases while of iso-paraffins increases, revealing that isomerization reactions are favoured by temperature.

The total liquid product of WCO catalytic hydrotreatment was further investigated in terms of its percentage that contains paraffins within the diesel boiling point range (220-360°C) [24]. The properties of WCO, hydrotreated WCO (total liquid product) and the diesel fraction of the hydrotreated WCO are presented in Table 3. Based on this study the overall yield of the WCO catalytic hydrotreatment technology was estimated over 92%v/v. The properties of the new 2nd generation paraffinic diesel product indicated a high-quality diesel with high heating value (49MJ/kg) and high cetane index (77) which is double of the one of fossil diesel. An additional advantage of the new biodiesel is its oxidation stability (exceeding 22hrs) and negligible acidity, rendering it as a safe biofuel, suitable for use in all engines. The properties and potential of the new biodiesel were further studied [25], for evaluating different fractions of the total liquid product and their suitability as an alternative diesel fuel.

 WCO Hydrotreated WCO Final biodiesel Density gr/cm3 0.896 0.7562 0.7869 C wt% 76.74 84.59 86.67 H wt% 11.61 15.02 14.74 S wppm 38 11.80 1.54 N wppm 47.42 0.77 1.37 O wt% 14.57 0.38 0 Recovery 0% °C 431.6 195.6 234.1 Recovery 10% °C 556.4 287.4 294.1 Recovery 30% °C 599 304.0 296.8 Recovery 50% °C 603.2 314.4 298.3 Recovery 70% °C 609 319.0 300 Recovery 90% °C 612.4 320.4 298.3 Recovery 100% °C 727.2 475.4 306.2

### Table 3.

Basic properties of waste cooking oil, hydrotreated waste cooking oil and final biodiesel

### 3.3. Pyrolysis oil upgrading

Pyrolysis oil is the product of fast pyrolysis of biomass, a process that allows the decomposition of large organic compounds of biomass such as lignin at medium temperatures in the presence of oxygen. Pyrolysis, that is in essence thermal cracking of biomass, is a well established process for producing bio-oil, the quality of which however is far too poor for direct use as transportation fuel. The product yields and chemical composition of pyrolysis oils depend on the biomass type and size as well as on the operating parameters of the fast pyrolysis. However, a major distinction between pyrolysis oils is based on whether catalyst is employed for the fast pyrolysis reactions or not. Non-catalytic pyrolysis oils have a higher water content than catalytic pyrolysis oils, rendering the downstream upgrading process a more challenging one for the case of the non-catalytic pyrolysis oils.

Untreated pyrolysis oil is a dark brown, free-flowing liquid with about 20-30% water that cannot be easily separated. It is a complex mixture of oxygenated compounds including water solubles (acids, alcohols, ethers) and water insolubles (n-hexane, di-chloor-methane), which is unstable in long-term storage and is not miscible with conventional hydrocarbon-based fuels. It should be noted that due to its nature pyrolysis oil can be employed for the production of a wide range of chemicals and solvents. However, if pyrolysis oil is to be used as a fuel for heating or transportation, it requires upgrading leading to its stabilization and conversion to a conventional hydrocarbon fuel by removing the oxygen through catalytic hydrotreating. For this reason, a lot of research effort is focused on catalytic hydrotreating of pyrolysis oil, as it is a process enabling oxygen removal and conversion of the highly corrosive oxygen compounds into aromatic and paraffinic hydrocarbons.

For non-catalytic pyrolysis oils, the catalytic hydrotreating upgrading process involves contact of pyrolysis oil molecules with hydrogen under pressure and at moderate temperatures (<400°C) over fixed bed catalytic reactors. Single-stage hydrotreating has proved to be difficult, producing a heavy, tar-like product. Dual-stage processing, where mild hydrotreating is followed by more severe hydrotreating has been found to overcome the reactivity of the bio-oil. Overall, the pyrolysis oil is almost completely deoxygenated by a combination of hydro deoxygenation and decarboxylation. In fact less than 2% oxygen remains in the treated, stable oil, while water and off-gas are also produced as byproducts. The water phase contains some dissolved organics, while the off-gas contains light hydrocarbons, excess hydrogen, and carbon dioxide. Once the stabilized oil is produced it can be further processed into conventional fuels or sent to a refinery. Table 1 shows the properties of some common catalytic pyrolysis oils according to literature.

Catalytic pyrolysis oils have been reported to getting upgraded via single step hydroprocessing, most of the times utilizing conventional CoMo and NiMo catalysts. During the single step hydroprocessing, the catalytic pyrolysis oil feedstock is pumped to high pressure, then mixed with compressed hydrogen and enters the hydroprocessing reactor. In Table 5 the typical operating parameters for single stage hydroprocessing and associated deoxygenation achievements are given according to literature [29;33-38].

 Types of Pyrolysis Biooils Properties Test Methods [26] [27] [28] [29] [30] [31] [32] H2O content (%wt) KarlFisher 20 23.9 30 20-30 29.85 pH pHmeter 2.2 2.5 2~3 2.5 Density 15C (Kg/L) ASTM D4052 1.207 1.2 1.15-1.2 1.192 1.2 1.19 HHV (MJ/Kg) DIN51900 17.57 LHV (MJ/Kg) DIN51900 15.83 Solids Content (%wt) Insolubles in Ethanol 0.06 Ash content (%wt) ASTM D482 0.0034 <0.1 0.1 0.15 0-0.2 Pour point ASTM D97 -30 -30 Flash point ASTM D93 48 40-65 40-65 51 Viscosity (cP) @ 40C 40 40-100 40-100 43-1510 Viscosity 20°C 9mm2/s) ASTM D445 47.18 Viscosity 50°C (mm2/s) ASTM D445 9.726 Carbon (%wt) ASTM D5291 42.64 40.1 51.1 ~52 39.17 54-58 39.4-46.7 Hydrogen (wt%) ASTM D5291 5.83 7.6 7.3 ~6.4 8.04 5.5-7 7.2-7.9 Nitrogen (wt%) ASTM D5291 0.1 0.1 ~0.2 0.05 0-0.2 0.2 Sulphur (%wt) ASTM 0.01 0.032 Clorine (%wt) ASTM 0.012 AlkaliMetals (%wt) ICP <0.003 Oxygen (wt%) 52.1 41.6 ~40 52.74 35-40 45.7-52.7

### Table 4.

Properties of different pyrolysis oils according to literature

 Catalyst CoMo [29][33][34][35][36], NiMo [34][35][36], others [37][38] Temperature (ºC) 350-420 Pressure (psig) 1450-2900 LHSV (Hr-1) 0.1-1.2 Deoxygenation (wt%) 78-99.9 Density (kg/l) 0.9-1.03

### Table 5.

Single-stage pyrolysis oil hydroprocessing operating parameters

However, in the case of non-catalytic pyrolysis oils or for achieving better quality products, multiple-stage hydroprocessing can be employed for upgrading pyrolysis oils. Multiple-stage hydroprocessing utilizes at least two different stages of hydroprocessing, which may include hydrotreating or hydrotreating and hydrocracking reactions. In the first stage the catalytic hydrotreatment reactor stabilizes the pyrolysis oil by mild hydrotreatment over CoMo or NiMo hydrotreating catalyst [32;40-42]. The first stage product is then further processed in the second-stage hydrotreater, which operates at higher temperatures and lower space velocities than the first stage hydrotreater, employing also CoMo or NiMo catalysts within the reactor. The 2nd stage product is separated into an organic-phase product, wastewater, and off-gas streams. In the literature [41], even a 3rdstage hydroprocessing has been used for the heavy fraction (which boils above 350°C) of the 2ndstage product, where hydrocracking reactions take place for converting the heavy product molecules into gasoline and diesel blend components.

 Feed 1st stage 2nd stage 3rd stage Catalyst CoMo[32][40],NiMo[32][42], others[39] CoMo[32][40]NiMo[32][42], others [39] CoMo[4141] Temperature (C°) 150-240 225-370 350-427 Pressure (psig) 1000-2000 2015 1280 LHSV (hr-1) 0.28-1 0.05-0.14 Deoxygenation (wt%) 60-98.6

### Table 6.

Multiple-step pyrolysis oil hydroprocessing operating parameters

### 3.4. Fischer-Tropsch wax upgrading

Biofuels production via the Fischer-Tropsch technology is a conversion process of solid biomass into liquid fuels (Biomass-To-Liquid or BTL) as it is depicted in Figure 2. More specifically the solid biomass is gasified in the presence of air and the produced biogas rich in CO and H2 (synthesis gas), after being pretreated to remove coke residues and sulfur compounds, enters the Fischer-Tropsch reactor. The Fischer-Tropsch reactions allow the catalytic conversion of the synthesis gas into a mixture of paraffinic hydrocarbons consisting of light (C1-C4), naphtha (C5-C11), diesel (C12-C20) and heavier hydrocarbons (>C20). Even though the Fischer-Tropsch reactions yields depend on the catalyst and operating parameters employed [43-45], the liquid product (naphtha, diesel and heavier hydrocarbons) yield is high (~95%). The produced synthetic naphtha and diesel fuels can be used similarly to their fossil counterparts. The heavier product however, which is called as Fischer-Tropsch wax, due to its waxy/paraffinic nature should get upgraded via catalytic hydrocracking to get converted to mid-distillate fuels (naphtha and diesel).

The conversion of Fischer-Tropsch wax into mainly diesel was studied in virtue of the European Project RENEW [46]. During this project Fischer-Tropsch wax with high paraffinic content of C20-C45 was converted into a total liquid product consisting of naphtha, kerosene and diesel fractions via catalytic hydrocracking. However the total liquid product content of diesel molecules was the highest and the diesel fraction was further separated and characterized having density of 0.78gr/ml and cetane index of 76 [47]. The schematic of the BTL process with actual images of the feedstock, Fischer-Tropsch wax and synthetic diesel are given in Figure 8.

### 3.5. Micro-algal oil conversion to biofuels

The rapid development of the biofuels production technologies from different biomass types has given rise to the biomass and food markets as it was aforementioned. Besides the use of residual biomass, research and in particular biotechnology has moved forward into seeking alternative biomass production technologies that will offer higher yields per hectare as well as lipids and carbohydrates, which are not part of the human and animal food-chain, avoiding competition between food/feed and energy crops. Targeted research efforts have offered a promising solution by the selection of unicellular microorganisms for the production of biofuels [48][49]. Micro-algae are photosynthetic microorganisms that can produce lipids, proteins and carbohydrates in large amounts over short periods of time.

Micro-algae are currently considered a prominent source of fatty acids, which offers large yields per hectare with various fatty acid foot-prints from each strain. In fact, there are certain strains that offer fatty acids of increased saturation (small content of unsaturated fatty acids) and of smaller carbon-chain length such as Dunaliellasalina, Chlorella minutissima, Spirulina maxima, Synechococcus sp.[50] etc. Another advantage of algal oils is that their fatty acid content can be directed to small carbon-chain molecules either genetically or by manipulating the aquaculture conditions such as light source and intensity [51], nitrogen starvation period [52], nutrients and CO2 feeding profiles [53].

Micro-algae and their products formulated the so called 3rd generation biofuels, as they incorporate various characteristics, which render them superior over other biofuels and biomass types. Micro-algae can also be produced in sea water [54] or even waste water, while they are biodegradable and relatively harmless during an eventual spill. Furthermore, their yield per hectare can reach 3785-5678lit, which is 20-700 higher over the conventional energy crops yield (soy, rape and palm). The lipids contained in most micro-algal oils have a similar synthesis with that of soy-bean oil, while they also contain some poly-saturated fatty acids with four double bonds. As a result catalytic hydrotreating of micro-algal oil is the most promising technology for converting it into biofuels.

### 3.6. Co-hydroprocessing

The effectiveness of catalytic hydroprocessing was also explored for co-processing of lipid feedstocks with petroleum fractions as catalytic hydroprocessing units are available in almost all refineries. The first co-processing study involved experiments of catalytic hydrotreating of sunflower oil mixtures with heavy petroleum fractions aiming to produce high quality diesel [55]. The experiments were conducted in a continuous fixed-bed reactor over a wide range of temperatures 300-450°C employing a typical NiMo/Al2O3 hydrotreating catalyst. The study was focused on the hydrogenation of double C-C bonds and the subsequent paraffin formation via the three different reactions routes: decarbonylation, decarboxylation and deoxygenation. Furthermore the large carbon-chain paraffins can also undergo isomerization and cracking leading to the formation of smaller paraffins. This study concluded that the selectivity of products on decarboxylation and decarbonylation is increasing as the temperature and vegetable oil content in the feedstock increase [55].

In a similar study catalytic hydrocracking over sunflower oil and heavy vacuum gas oil mixtures was investigated [56]. The experiments were conducted in a continuous-flow hydroprocessing pilot-plant over a range of temperatures (350-390°) and pressures (70-140bar). Three different hydrocracking catalysts were compared under the same conditions and four different feedstocks were employed, incorporating for 10% and 30%v/v of lipid bio-based feedstock and considering non-pretreated and pretreated sunflower oil as a bio-based feedstock. The results indicated that a prior mild hydrogenation step of sunflower oil is necessary before hydrocracking. Furthermore, conversion was increased with increasing sunflower oil ratio in the feedstock and increasing temperature, while the later decreased diesel selectivity.

The effect of the process parameters and the vegetable oil content of the feedstocks on the yield, physical properties, chemical properties and application properties during co-hydrotreating of sunflower oil and gas-oil mixtures utilizing a typical NiMo/Al2O3hydrotreating catalyst was also studied [57]. The experimental results of this study indicated that catalytic co-hydrogenation of gas oil containing sunflower oil in different percentages allowed both vegetable oil conversion reactions (saturation, deoxygenation) and the gas oil quality improvement reactions (hetero atom removal, aromatic reduction). The optimal operating conditions (360-380°C, P=80 bar, LHSV=1.0h-1, H2/oil=600 Nm3/m3and 15% sunflower oil content of feed) resulted in a final diesel product with favorable properties (e.g. less than 10 wppm sulfur, ~20% aromatics) but poor cold flow properties (CFPP=3°C). The study also indicated that for sunflower content in the feedstock higher than 15% reduced the desulfurization efficiency. Furthermore, the authors also concluded that the presence of sunflower oil in the feedstock has augmented the normal and iso-paraffins content of the final product and as a result has increased the cetane number but degraded the cold flow properties, indicating that an isomerization step is required as an additional step.

The issue of catalyst development suitable for co-hydrotreating and co-hydrocracking of gas-oil and vegetable oil mixtures was recently addressed [10], as there are no commercial hydroprocessing catalysts available for lipid feedstocks. New sulfided Ni–W/SiO2–Al2O3 and sulfided Ni–Mo/Al2O3 catalysts were tested for hydrocracking and hydrotreating of gas-oil and vegetable oil mixtures respectively. The results indicated that the hydrocracking catalyst was more selective for the kerosene hydrocarbons (140–250°C), while the less acidic hydrotreating catalyst was more selective for the diesel hydrocarbons (250–380°C). The study additionally showed that the deoxygenation reactions are more favored over the hydrotreating catalyst, while the decarboxylation and decarbonylation reactions are favored over the hydrocracking catalyst.

## 4. Demo and industrial applications

As catalytic hydrotreating of liquid biomass has given promising results, the industrial world has given enough confidence to apply it in pilot and industrial scale. The NesteOil Corporation has developed the NExBTL technology for converting vegetable oil (primarily palm oil) into a renewable diesel also known as “green” diesel (Figure 9). Based on this technology the first catalytic hydrotreatment of vegetable oils unit was constructed in Finland in 2007, within the existing Poorvo refinery of NesteOil, with a capacity of 170 kton/hr. The primary feedstock is palm oil, while it can also process rapeseed oil and even waste cooking oil. The same company has constructed a second unit within the same refinery while it has also planned to construct two new units, one in Singapore and one in Rotterdam, with the capacity of 800 kton/yr each.

The catalytic hydrotreatment technology of 100% waste cooking oil for biodiesel production was developed in the Centre for Research and Technology Hellas (CERTH) in Thessaloniki, Greece [21-24] and later demonstrated via the BIOFUELS-2G project [59], which was co-funded by the European Program LIFE”. In this project WCO was collected from associated restaurants and the produced 2nd generation bio-diesel, to be called “white diesel” was employed. For the demonstration of the new technology, 2 tons of “white diesel’ were produced via catalytic hydrotreatment of WCO based on the large-scale pilot units available in CERTH. The production process simplified diagram is given in Figure 10. The new fuel will be applied to a garbage truck in a 50-50 mixture with conventional diesel in August 2012, aiming to promote the new technology as it exhibits overall yields exceeding 92% v/v.

In the USA the Dynamic Fuels company [60] has constructed in Baton Rouge a catalytic hydrotreating unit dedicated to oils and animal fats with 285 Mlit capacity. The unit employs the Syntroleum technology based on Fischer-Tropsch for the production of synthetic 2nd generation Biodiesel while it also produces bio-naphtha and bio-LPG. The Bio-Synfining technology of Syntroleum converts the triglycerides of fats and oils into n- and iso-paraffins via catalytic hydrogenation, thermal cracking and isomerization as it is applied in the Fischer-Tropsch wax upgrading to renewable diesel (R-2) and renewable jet (R-8) fuel.

## 5. Future perspectives

Catalytic hydrotreating of liquid biomass is continuously gaining ground as the most effective technology for liquid biomass conversion to both ground- and air-transportation fuels. The UOP company of Honeywell, via the technology it has developed for catalytic hydrotreating of liquid biomass (Figure 11), has announced imminent collaboration with oil and airline companies such as Petrochina, Air China and Boeing for the demonstration of the sustainable air-transport in China. This initiative will lead a strategic collaboration between the National Energy Agency of china with the Commerce and Development Agency of USA leading to the development of the new biofuels market in China.

In the EU airline companies collaborate with universities, research centers and biofuels companies in order to confront their extensive contribution to CO2 emissions. Since 2008 most airline companies promote the use of biofuels in selected flights as shown in Table 7 [62]. As it is obvious most pilot flights have taken place with Hydrotreated Renewable Jet (HRJ), which is kerosene/jet produced via catalytic hydrotreatment of liquid biomass. Moreover, Lufthansa has also completed a 6-month exploration program of employing HRJ in a 50/50 mixture with fossil kerosene in one of the 4 cylinders of a plane employed for the flight between Hamburg-Frankfurt-Hamburg with excellent results [63].

Besides the future applications for air-transportation, the automotive industry is also exhibiting increased interest for the broad use of biofuels resulting from catalytic hydrotreatment of liquid biomass. In fact these paraffinic biofuels can be employed in higher than 7%v/v blending ratio (which is the maximum limit for FAME) as they exhibit high cetane number and have significant oxidation stability [64]

 Airline Aircraft Partners Biofuel(lipid sources) Blend* Virgin Atlantic B747-400 Boeing, GE Aviation FAME(coconut & palm) 20% Air New Zealand B747-400 Boeing, Rolls-Royce HRJ(Jatropha) 50% Contintental Airlines B737-800 Boeing, GE Aviation, CFM, Honeywell UOP HRJ(Jatropha&algea) 50% JAL B747-300 Boeing, Pratt&Whitney, Honeywell UOP HRJ(Camelina, Jatropha& algae) 50% KLM B747-400 GE, Honeywell UOP HRJ(Camelina) 50% TAM A320 Airbus, CFM HRJ(Jatropha) 50%

### Table 7.

Pilot flights with biofuels [62]

The highest interest is exhibited by oil companies around the catalytic hydrotreatment of liquid biomass technology for the production of biofuels and particularly to its application to oil from micro-algae. ExxonMobil has invested 600M$in the Synthetic Genomics company of the pioneer scientist Craig Ventner aiming to research of converting micro-algae to biofuels with minimal cost. BP has also invested 10M$ for collaboration with Martek for the production of biofuels from micro-algae for air-, train-, ground- and marine transportation applications.

## 6. Conclusion

Catalytic hydrotreatment of liquid biomass is the only proven technology that can overcome its limitations as a feedstock for fuel production (low H/C ratio, high oxygen and water content). Even though it has recently started to be investigated as an alternative technology for biofuels production, it fastly gains ground due to the encouraging experimental results and successful pilot/demo and industrial applications. Catalytic hydrotreatment of liquid biomass leads to a wide range of new alterative fuels including bio-naphtha, bio-jet and bio-diesel, are paraffinic in nature and as a result exhibiting high heating values, increased oxidation stability and negligible acidity and corrosivity. As a result it is not over-optimistic to claim that this technology will broaden the biofuels market into scales capable to actually mitigate the climate change problems.

## Acknowledgements

The author would like to thank Ms Iva Simcic and InTech Europe for enabling her to publish this book chapter, while she is grateful to Mr Athanasios Dimitriadis who provided support, offered comments, proofreading and design. Finally she would like to express her appreciation for the financial support provided by the EU project BIOFUELS-2G which is co-financed by the European Program LIFE+.

## How to cite and reference

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Stella Bezergianni (March 20th 2013). Catalytic Hydroprocessing of Liquid Biomass for Biofuels Production, Liquid, Gaseous and Solid Biofuels, Zhen Fang, IntechOpen, DOI: 10.5772/52649. Available from:

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Here we will solve some numerical problems on the properties of isosceles triangles Find x° from the given figures. In ∆XYZ, XY = XZ. Therefore, ∠XYZ = ∠XZY = x°. Now, ∠YXZ + ∠XYZ + XZY = 180° ⟹ 84° + x° + x° = 180° ⟹ 2x° = 180° - 84° ⟹ 2x° = 96°

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### Three Angles of an Equilateral Triangle are Equal | Axis of Symmetry

Here we will prove that if the three angles of a triangle are equal, it is an equilateral triangle. Given: In ∆XYZ, ∠YXZ = ∠XYZ = ∠XZY. To prove: XY = YZ = ZX. Proof: Statement 1. XY = ZX. 2. XY = YZ. 3. XY = YZ = ZX. (Proved) Reason 1. Sides opposite to equal angles ∠XZY

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### Sides Opposite to the Equal Angles of a Triangle are Equal | Diagram

Here we will prove that the sides opposite to the equal angles of a triangle are equal. Given: In ∆ABC, ∠XYZ = ∠XZY. To prove: XY = XZ. Construction: Draw the bisector XM of ∠YXZ so that it meets YZ at M. Proof: Statement 1. In ∆XYM and ∆XZM, (i) ∠XYM = XZM (ii) ∠YXM = ∠ZXM

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### The Three Angles of an Equilateral Triangle are Equal | With Diagram

Here we will prove that the three angles of an equilateral triangle are equal. Given: PQR is an equilateral triangle. To prove: ∠QPR = ∠PQR = ∠ PRQ. Proof: Statement 1. ∠QPR = ∠PQR 2. ∠PQR = ∠ PRQ. 3. ∠QPR = ∠PQR = ∠ PRQ. (Proved). Reason 1. Angles opposite to equal sides QR

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### Equal Sides of an Isosceles Triangle are Produced, the Exterior Angles

Here we will prove if the equal sides of an isosceles triangle are produced, the exterior angles are equal. Given: In the isosceles triangle PQR, the equal sides PQ and PR are produced to S and T respectively. To prove: ∠RQS = ∠QRT. Proof: Statement 1. ∠PQR = ∠PRQ 2. ∠RQS

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### Angles Opposite to Equal Sides of an Isosceles Triangle are Equal

Here we will prove that in an isosceles triangle, the angles opposite to the equal sides are equal. Solution: Given: In the isosceles ∆XYZ, XY = XZ. To prove ∠XYZ = ∠XZY. Construction: Draw a line XM such that it bisects ∠YXZ and meets the side YZ at M. Proof: Statement

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### Application of Congruency of Triangles | Isosceles triangle Proved

Here we will prove some Application of congruency of triangles. PQRS is a rectangle and POQ an equilateral triangle. Prove that SRO is an isosceles triangle. Solution: Given: PQRS is a rectangle. POQ is an equilateral triangle to prove ∆SOR is an isosceles triangle. Proof:

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### Prove that the Bisectors of the Angles of a Triangle Meet at a Point

Here we will prove that the bisectors of the angles of a triangle meet at a point. Solution: Given In ∆XYZ, XO and YO bisect ∠YXZ and ∠XYZ respectively. To prove: OZ bisects ∠XZY. Construction: Draw OA ⊥ YZ, OB ⊥ XZ and OC ⊥ XY. Proof: Statement 1. In ∆XOC and ∆XOB

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### Point on the Bisector of an Angle | Corresponding Parts of a Triangles

Here we will prove that any point on the bisector of an angle is equidistant from the arms of that angle. Solution: Given OZ bisects ∠XOY and PM ⊥ XO and PN ⊥ OY. To prove PM = PN. Proof: Statement 1. In ∆OPM and ∆OPN, (i) ∠MOP = ∠NOP. (ii) ∠OMP = ∠ONP = 90°

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### Prove that an Altitude of an Equilateral Triangle is also a Median

Here we will prove that an altitude of an equilateral triangle is also a median. In a ∆PQR, PQ = PR. Prove that the altitude PS is also a medina. Solution: Given in ∆PQR, PQ = PR and PS ⊥ QR.To prove PS is a median, i.e., QS = SR Proof: Statement 1. In ∆PQS and ∆PRS,

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### Problems on Congruency of Triangles |Prove Two Triangles are Congruent

Here we will learn how to prove different types of problems on congruency of triangles. 1. PQR and XYZ are two triangles in which PQ = XY and ∠PRQ = 70, ∠PQR = 50°, ∠XYZ = 70°, and ∠YXZ = 60°. Prove that the two triangles are congruent. Solution: In a triangle, the sum of

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### Criteria for Congruency | SAS| AAS | SSS | RHS | CPCTC

Here we will learn different criteria for congruency of triangles. I. SAS (Side-Angle-Side) Criterion: If two triangles have two sides of one equal to two sides of the other, each to each, and the angles included by those sides are equal then the triangles are congruent.

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