DIMENSIONAL HOMOGENEITY
We’ve all heard the old saying, You can’t add apples and oranges (Fig. 7–3).
This is actually a simplified expression of a far more global and fundamental
mathematical law for equations, the law of dimensional homogeneity,
stated as
Every additive term in an equation must have the same dimensions.
Consider, for example, the change in total energy of a simple compressible
closed system from one state and/or time (1) to another (2), as illustrated in
Fig. 7–4. The change in total energy of the system ( E) is given by Change of total energy of a system:


where E has three components: internal energy (U), kinetic energy (KE), and potential energy (PE). These components can be written in terms of the system mass (m); measurable quantities and thermodynamic properties at each of the two states, such as speed (V), elevation (z), and specific internal
energy (u); and the known gravitational acceleration constant (g),

It is straightforward to verify that the left side of Eq. 7–2 and all three additive terms on the right side of Eq.  have the same dimensions—energy. Using the definitions of Eq. 7–3, we write the primary dimensions of each term,
 If at some stage of an analysis we find ourselves in a position in which two additive terms in an equation have different dimensions, this would be a clear indication that we have made an error at some earlier stage in the analysis . In addition to dimensional homogeneity, calculations are valid only when the units are also homogeneous in each additive term. For example, units of energy in the above terms may be J, N · m, or kg ·
m2/s2, all of which are equivalent. Suppose, however, that kJ were used in place of J for one of the terms. This term would be off by a factor of 1000 compared to the other terms. It is wise to write out all units when performing mathematical calculations in order to avoid such errors.
EXAMPLE : Dimensional Homogeneity of the Bernoulli Equation Probably the most well-known (and most misused) equation in fluid mechanics is the Bernoulli equation (Fig. 7–6), discussed in Chap. 5. The standard form of the Bernoulli equation for incompressible irrotational fluid flow is :
(a) Verify that each additive term in the Bernoulli equation has the same dimensions. (b) What are the dimensions of the constant C?
SOLUTION We are to verify that the primary dimensions of each additive term in Eq. 1 are the same, and we are to determine the dimensions of constant C.
Analysis (a) Each term is written in terms of primary dimensions,
Primary dimensions of the Bernoulli constant:

Discussion If the dimensions of any of the terms were different from the
others, it would indicate that an error was made somewhere in the analysis.
Nondimensionalization of Equations The law of dimensional homogeneity guarantees that every additive term in an equation has the same dimensions. It follows that if we divide each term in the equation by a collection of variables and constants whose product has those same dimensions, the equation is rendered nondimensional (Fig.). If, in addition, the nondimensional terms in the equation are of order unity, the equation is called normalized. Normalization is thus more restrictive than nondimensionalization, even though the two terms are sometimes (incorrectly) used interchangeably. Each term in a nondimensional equation is dimensionless. In the process of nondimensionalizing an equation of motion, nondimensional parameters often appear—most of which are named after a notable scientist or engineer (e.g., the Reynolds number and the Froude number). This process is referred to by some authors as inspectional analysis.
Dimensional variables are defined as dimensional quantities that change or vary in the problem. For the simple differential equation given in Eq.  there are two dimensional variables: z (dimension of length) and t (dimension of time). Nondimensional (or dimensionless) variables are defined as quantities that change or vary in the problem, but have no dimensions; an example is angle of rotation, measured in degrees or radians which are dimensionless units. Gravitational constant g, while dimensional, remains constant and is called a dimensional constant. Two additional dimensional constants are relevant to this particular problem, initial location z0 and initial vertical speed w0. While dimensional constants may change from problem to problem, they are fixed for a particular problem and are thus distinguished from dimensional variables. We use the term parameters for the combined set of dimensional variables, nondimensional variables, and dimensional constants in the problem. Equation* is easily solved by integrating twice and applying the initial conditions. The result is an expression for elevation z at any time t:

The constant and the exponent 2 in Eq.  are dimensionless results of the integration. Such constants are called pure constants. Other common examples of pure constants are p and e. To nondimensionalize Eq.  we need to select scaling parameters, based on the primary dimensions contained in the original equation. In fluid flow problems there are typically at least three scaling parameters, e.g., L, V, and P0 P , since there are at least three primary dimensions in the general problem (e.g., mass, length, and time). In the case of the falling object being discussed here, there are only two primary dimensions, length and time, and thus we are limited to selecting only two scaling parameters. We have some options in the selection of the scaling parameters since we have three available dimensional constants g, z0, and w0. We choose z0 and w0. You are invited to repeat the analysis with g and z0 and/or with g and w0. With these two chosen scaling parameters we nondimensionalize the dimensional variables z and t. The first step is to list the primary dimensions of all dimensional variables and dimensional constants in the problem, Primary dimensions of all parameters:
The second step is to use our two scaling parameters to nondimensionalize z and t (by inspection) into nondimensional variables z* and t*, Nondimensionalized variables:
Substitution :

 which is the desired nondimensional equation. The grouping of dimensional constants in Eq. 7–7 is the square of a well-known nondimensional parameter or dimensionless group called the Froude number,
The Froude number also appears as a nondimensional parameter in freesurface flows, and can be thought of as the ratio of inertial force to gravitational force (Fig.). You should note that in some older textbooks, Fr is defined as the square of the parameter shown in Eq.. Substitution of Eq into Eq yields Nondimensionalized equation of motion:
In dimensionless form, only one parameter remains, namely the Froude number. Equation 7–9 is easily solved by integrating twice and applying the initial conditions. The result is an expression for dimensionless elevation z* at any dimensionless time t*:
Nondimensional result:
Comparison of Eqs. 5 and 10 reveals that they are equivalent. In fact, for practice, substitute Eqs. 6 and 7–8 into Eq. 5 to verify Eq. 10. It seems that we went through a lot of extra algebra to generate the same final result. What then is the advantage of nondimensionalizing the equation? Before answering this question, we note that the advantages are not so clear in this simple example because we were able to analytically integrate the differential equation of motion. In more complicated problems, the differential equation (or more generally the coupled set of differential equations) cannot be integrated analytically, and engineers must either integrate the equations numerically, or design and conduct physical experiments to obtain the needed results, both of which can incur considerable time and expense. In such cases, the nondimensional parameters generated by nondimensionalizing the equations are extremely useful and can save much effort and expense in the long run.
There are two key advantages of nondimensionalization . First, it increases our insight about the relationships between key parameters. Equation 8 reveals, for example, that doubling w0 has the same effect as decreasing z0 by a factor of 4. Second, it reduces the number of parameters in the problem. For example, the original problem contains one dependent variable, z; one independent variable, t; and three additional dimensional constants, g, w0, and z0. The nondimensionalized problem contains one dependent parameter, z*; one independent parameter, t*; and only one additional parameter, namely the dimensionless Froude number, Fr. The number of additional parameters has been reduced from three to one!
Example (next) : further illustrates the advantages of nondimensionalization.
Illustration of the Advantages of Nondimensionalization
Your little brother’s high school physics class conducts experiments in a large vertical pipe whose inside is kept under vacuum conditions. The students are able to remotely release a steel ball at initial height z0 between 0 and 15 m (measured from the bottom of the pipe), and with initial vertical speed w0 between 0 and 10 m/s. A computer coupled to a network of photosensors along the pipe enables students to plot the trajectory of the steel ball (height z plotted as a function of time t) for each test. The students are unfamiliar with dimensional analysis or nondimensionalization techniques, and therefore conduct several “brute force” experiments to determine how the trajectory is affected by initial conditions z0 and w0. First they hold w0 fixed at 4 m/s and conduct experiments at five different values of z0: 3, 6, 9, 12, and 15 m. The experimental results are shown in Fig. 12a. Next, they hold z0 fixed at 10 m and conduct experiments at five different values of w0: 2, 4, 6, 8, and 10 m/s. These results are shown in Fig. 12b. Later that evening, your brother shows you the data and the trajectory plots and tells you that they plan to conduct more experiments at different values of z0 and w0. You explain to him that by first nondimensionalizing the data, the problem can be reduced to just one parameter, and no further experiments are required. Prepare a nondimensional plot to prove your point and discuss.
SOLUTION A nondimensional plot is to be generated from all the available trajectory data. Specifically, we are to plot z* as a function of t*.Assumptions The inside of the pipe is subjected to strong enough vacuum pressure that aerodynamic drag on the ball is negligible. Properties The gravitational constant is 9.81 m/s2. Analysis Equation 4 is valid for this problem, as is the nondimensionalizatio that resulted in Eq. 9. As previously discussed, this problem combines three of the original dimensional parameters (g, z0, and w0) into one nondimensional parameter, the Froude number. After converting to the dimensionless variables of Eq. 6, the 10 trajectories of Fig. 12a and b are replotted in dimensionless format in Fig. 13. It is clear that all the
trajectories are of the same family, with the Froude number as the only remaining parameter. Fr2 varies from about 0.041 to about 1.0 in these experiments. If any more experiments are to be conducted, they should include combinations of z0 and w0 that produce Froude numbers outside of this range. A large number of additional experiments would be unnecessary, since all the trajectories would be of the same family as those plotted in Fig. 13. Discussion At low Froude numbers, gravitational forces are much larger than inertial forces, and the ball falls to the floor in a relatively short time. At large values of Fr on the other hand, inertial forces dominate initially, and the ball rises a significant distance before falling; it takes much longer for the ball to hit the ground. The students are obviously not able to adjust the gravitational constant, but if they could, the brute force method would require many more experiments to document the effect of g. If they nondimensionalize first, however, the dimensionless trajectory plots already obtained and shown in Fig.13 would be valid for any value of g; no further experiments would be required unless Fr were outside the range of tested values. If you are still not convinced that nondimensionalizing the equations and the parameters has many advantages, consider this: In order to reasonably document the trajectories of Example 3 for a range of all three of the dimensional parameters g, z0, and w0, the brute force method would require several (say a
minimum of four) additional plots like Fig. 12a at various values (levels) of w0, plus several additional sets of such plots for a range of g. A complete data set for three parameters with five levels of each parameter would require 53 125 experiments! Nondimensionalization reduces the number of parameters from three to one—a total of only 51 5 experiments are required for the same resolution. (For five levels, only five dimensionless trajectories like those of Fig. are required, at carefully chosen values of Fr.) Another advantage of nondimensionalization is that extrapolation to untested values of one or more of the dimensional parameters is possible. For example, the data of Example 3 were taken at only one value of gravitational acceleration. Suppose you wanted to extrapolate these data to a different value of g.

in this topic, we first review the concepts of dimensions and units. We then review the fundamental principle of dimensional homogeneity, and show how it is applied to equations in order to nondimensionalize them and to identify dimensionless groups. We discuss the concept of similarity between a model and a prototype. We also describe a powerful tool for engineers and scientists called dimensional analysis, in which the combination of dimensional variables, nondimensional variables, and dimensional constants into nondimensional parameters reduces the number of necessary independent parameters in a problem. We present a step-by-step method for obtaining these nondimensional parameters, called the method of repeating variables, which is based solely on the dimensions of the variables and constants. Finally, we apply this technique to several practical problems to illustrate both its utility and its limitations.

DIMENSIONS AND UNITS
A dimension is a measure of a physical quantity (without numerical values), while a unit is a way to assign a number to that dimension. For example, length is a dimension that is measured in units such as microns ( m), feet (ft), centimeters (cm), meters (m), kilometers (km), etc. There are seven primary dimensions (also called fundamental or basic dimensions)—mass, length, time, temperature, electric current, amount of light, and amount of matter.
All nonprimary dimensions can be formed by some combination of the seven primary dimensions.
For example, force has the same dimensions as mass times acceleration (by
Newton’s second law). Thus, in terms of primary dimensions,
Dimensions of force:


where the brackets indicate “the dimensions of” and the abbreviations are taken from Table . You should be aware that some authors prefer force instead of mass as a primary dimension—we do not follow that practice.
 

Heat transfer is a complex process which is achieved by the superposition of three fundamental modes: conduction, convection and radiation. In case one of these three modes is determining the effects of others are unimportant, so they can be neglected, greatly simplifying the analysis cases concerned. Taking into account these conditions, we will then present the mechanism of occurrence of each of these modes.

 Conduction:

In areas where there is a material temperature gradient there is a mechanism of heat transfer from warm regions to cold regions that does not involve radiation or travel macroscopic matter is conduction.
In gases, the molecules that hot regions with more energy by collision with molecules of lower kinetic energy when they enter the colder regions. From a macroscopic point of view, it thus heat transfer. In liquids, the process is quite similar, but the molecules are closer, free electrons and lattice vibrations. It is thus understood that the body that are good conductors of electricity are also good conductors of heat. Similarly, good electrical insulators are good thermal insulators.
Heat output transmitted by conduction through a surface element dS located inside a material or temperature gradient is given by Fourier's law:

 Where k is thermal conductivity of the material (W / m ° K) and n is the unit vector normal to the worn by ds. The estimation of the thermal coefficient is generally impossible except by relation to which the gas kinetic theory can show that it must grow as the square root of the temperature. In liquids and solids, experience shows that k varied little with temperature.
The general equation for conduction is obtained by writing that the heat stored during the time dt by a small volume element dV material at temperature T is equal to the heat generated in the item minus the heat lost by conduction:

Every one (engineer) wants to get ahead in their career. We want more responsibility , more good work, prestige, better pay and benefits, and the corner office. That's fine and good. I'm with you. The challenge, then, is how do you actually make it happen? How do you go from wanting to achieving? In my experience, the best way to get what you want is simply to ask for it. 
Ask and you Shall Receive
I know it sounds crazy, but think about it for a minute - how are you going to get what you're looking for if the people who can make it happen don't know what you want? When you go to a restaurant, do you sit at an open table and just hope that the waiter will bring you the right meal? Heck, no! So why would you expect to get any kind of promotion or special assignment at the office if you don't show interest in them?
Some Examples
In my career, I've had a few instances where simply asking for a certain role has resulted in great things for me. During undergrad, I had a co-op gig with a circuit board manufacturing company. One day, the president held a meeting for all staff to announce that the company had won a grant to start a spin-off business to design LED streetlights. I thought that was pretty cool, so I went up to the president afterwords to chat about the new project and to see if he needed help with it. As it happens, they did! I ended up being the first full-time engineer on staff and was even offered stock options to stick with the company in it's early days. All because I just asked about it! Now, I ended up turning down the offer to follow a girl across the country, but that's another story....
I had a similar experience recently with my current employer during my annual performance review. I told my boss I was interested in getting into project management and to be more involved in the leadership of engineering teams as I move forward in my career. His response was to tell me he would support me in my getting my MBA in support of those goals. It's amazing how willing people are to help you towards your goals if you make it known what those goals are.
Some Steps to Get You Started

1. Decide what you want from your career - This is simple enough, right? How can you ask for support in developing yourself if you don't know what you need help achieving? This may be easier said that done, though. Maybe you're still in the early throws of your career, and you're still learning about everything an engineer can do in the fist place. That's OK, because you can ask for help figuring this out too! In my experience, if you demonstrate drive, people are very willing to help you with direction. People tend to enjoy guiding the motivated.

1. Decide who to ask - Think about who's job it is to help develop you in your career. In most organizations, this is probably your direct supervisor. Think about it this way: your boss' goal should be to promote all of their employees up to his or her level. If they can do that, then they demonstrate their abilities as a leader in their own right, and can earn themselves a promotion. Make no mistake about it, part of your boss' job is to make you better at what you do, and they're looking for ways to do that. Help them out, and good things can happen.

1. Decide when to ask - It might not be a fantastic idea to waltz into your manager's office randomly and proclaim "I want to be your boss some day!" If you have some kind of performance review on an annual or semi-annual basis, that would be a perfect opportunity to bring up where you'd like to go. You're already talking about where you've been in the past, why not talk about where you see yourself in the future?

Have you ever asked for something at work and were surprised that things worked out in your favour? Do you have any tips for other readers looking to do the same? Share your experience with us in the comments section below!


About the Author
Pat Sweet is an Electrical Engineer in Training in Ontario, Canada. He’s a full-time designer of all-electric trains and the author behind the Engineering and Leadership blog, where he shares his thoughts and experiences on leadership, productivity, and career advice for young engineers.