Divya Tyagi: Revolutionizing Wind Energy Through Mathematics

 

Divya Tyagi: Revolutionizing Wind Energy Through Mathematics

Divya Tyagi, an Indian-origin aerospace engineering graduate from Pennsylvania State University, has achieved a groundbreaking milestone in the field of aerodynamics and renewable energy. By revisiting and refining a century-old mathematical problem originally posed by British aerodynamicist Hermann Glauert, Tyagi has paved the way for significant advancements in wind turbine efficiency and design.

The Challenge: Revisiting Glauert's Equation

Hermann Glauert's 1926 work laid the foundation for understanding how wind turbines convert wind energy into electricity. His third-order polynomial equation focused on maximizing the power coefficient—a measure of efficiency—but left out critical factors such as:

• The total force and moment coefficients acting on rotor blades

• Structural challenges like blade bending under wind pressure

For decades, engineers worked within the limitations of this model. However, Tyagi's research identified gaps in Glauert's approach and sought to simplify and expand it using advanced mathematical techniques like the calculus of variations[1][2][3].

Tyagi's Breakthrough

Tyagi developed an addendum to Glauert's problem, enabling researchers to determine optimal aerodynamic performance for wind turbines. Her solution simplifies the complex mathematics while enhancing accuracy, allowing engineers to:

• Solve for ideal flow conditions to maximize power output

• Account for structural stresses like downwind thrust forces and root bending moments[1][4]

This refined model doesn't just improve turbine efficiency but also opens unexplored possibilities in design. According to her adviser Sven Schmitz, Tyagi's work will likely influence the next generation of wind turbines and become a staple in engineering classrooms worldwide[1][3].

Real-World Implications

The impact of Tyagi’s research extends beyond academic circles. A mere 1% improvement in a turbine’s power coefficient could significantly increase energy production, potentially powering entire neighborhoods more efficiently[3][4]. This advancement aligns with global efforts to transition toward sustainable energy solutions.

Recognition and Future Research

For her groundbreaking thesis, Tyagi received Penn State's prestigious Anthony E. Wolk Award for excellence in aerospace engineering research. Currently pursuing her master's degree, she is working on computational fluid dynamics (CFD) simulations funded by the U.S. Navy. Her ongoing project focuses on improving helicopter flight safety by analyzing airflow interactions between ships and helicopters during landing operations[3][4].

Example: How Tyagi’s Formula Boosts a Wind Turbine’s Performance

Scenario:
A wind turbine with a 100-meter rotor operates in 10 m/s average wind speed.

• Baseline (Glauert’s Model):

  • Power coefficient $C_P=48\mathrm{\%}$ (typical for modern turbines, below Betz’s 59.3% limit).

  • Annual Energy Production (AEP):

    $$\text{AEP}=0.5\times 1.225{\text{kg/m^}}^3\times \pi (50{)^}^2\times {\mathrm{10^}}^3\times 0.48\times 8760\text{hours }$$$$\approx 130,\text{\hspace{0.17em}⁣}000\text{MWh/year}$$

By solving her Euler-Lagrange equation, Tyagi’s model achieves:

• $C_P=49\mathrm{\%}$ (1% gain)

• Thrust coefficient $C_T$ reduced by 10% (lowering blade stress)

Results:

• New AEP:

  $$\mathrm{ \; \; \; 130},\text{\hspace{0.17em}⁣}000\text{MWh}\times 1.01=131,\text{\hspace{0.17em}⁣}300\text{MWh/year }$$

  → +1,300 MWh/year extra energy (powers ~120 more homes annually)

• Structural Benefit:
  Reduced thrust means blades experience 8–15% less bending force, potentially extending turbine lifespan by years.

Why It Matters:
For a wind farm with 50 turbines, Tyagi’s tweak generates 65,000 MWh more annual output—equivalent to saving 46,000 tons of CO₂ (vs. coal power) or earning $6.5M more revenue (at $100/MWh).

A Legacy of Persistence

Tyagi’s achievement is a testament to her determination and ingenuity. Her adviser Sven Schmitz remarked that she was the only student among four who successfully tackled this challenging problem. Despite spending countless hours on mathematically intensive research, Tyagi’s persistence has led to a solution that simplifies decades-old complexities while unlocking new opportunities in renewable energy[3][4].

Conclusion

Divya Tyagi’s work exemplifies how innovative thinking can bridge gaps between theoretical mathematics and practical applications. Her contributions not only advance wind energy technology but also inspire future researchers to challenge established norms and redefine possibilities in engineering.

Citations:

1. https://www.ndtv.com/world-news/indian-origin-student-at-us-university-solves-100-year-old-math-problem-7968018
2. https://timesofindia.indiatimes.com/education/news/meet-divya-tyagi-the-penn-state-student-who-cracked-a-100-year-old-wind-energy-equation-boosting-turbine-efficiency/articleshow/119260883.cms
3. https://www.sciencedaily.com/releases/2025/02/250226175933.htm
4. https://www.psu.edu/news/engineering/story/student-refines-100-year-old-math-problem-expanding-wind-energy-possibilities
5. https://mysilsila.com/blog/2025-03-18-Luminaries-Real-Influencers--How-Divya-Tyagi-Revived-a-100-Year-Old-Equation-to-Improve-Wind-Energy
6. https://pipeline.psu.edu/news/divya-tyagi-refines-100-year-old-math-problem-expanding-wind-energy-possibilities
7. https://www.thedailystar.net/tech-startup/news/university-student-refined-100-year-old-math-problem-3858206
8. https://collegementor.com/news/divya-tyagi-creates-history-solving-a-century-old-unsolved-math

Understanding the Carbon Footprint of Renewable Energy Sources

 Understanding the Carbon Footprint of Renewable Energy Sources

 As the world moves toward more sustainable energy solutions, understanding the environmental impact of renewable energy sources is crucial. While renewable energy is often promoted as "clean" or "zero-carbon," it's important to recognize that no energy source is completely free from environmental impact. This blog explores the carbon footprint of various renewable energy sources and compares their environmental impacts.

 

1. Solar Energy (Photovoltaic)

 

Solar panels are a cornerstone of modern renewable energy. They convert sunlight directly into electricity with minimal emissions during operation. However, the production of solar panels involves energy-intensive processes, leading to a carbon footprint of approximately 20-50 gCO₂e/kWh. Once installed, solar panels generate clean energy, making their overall impact relatively low compared to fossil fuels.

 

2. Solar Tower (Concentrated Solar Power)

Concentrated Solar Power (CSP) uses mirrors or lenses to concentrate sunlight onto a small area, generating heat that drives turbines to produce electricity. CSP systems have a carbon footprint of around 10-40 gCO₂e/kWh. The construction and maintenance of solar towers can be energy-intensive and require significant water resources for cooling. However, the operational emissions are minimal.

 

3. Biomass

Biomass energy involves burning organic materials like wood, agricultural residues, or dedicated energy crops to produce electricity. The carbon footprint of biomass can vary widely, ranging from 35-200 gCO₂e/kWh, depending on factors such as feedstock type and processing methods. Biomass can be carbon-neutral if the carbon dioxide emitted during combustion is offset by the carbon absorbed during plant growth. Nonetheless, unsustainable biomass practices can lead to deforestation, habitat loss, and increased emissions.

 

4. Geothermal Energy

Geothermal energy harnesses heat from the Earth's interior to generate electricity. This method has a relatively low carbon footprint of 5-50 gCO₂e/kWh. Geothermal systems produce minimal greenhouse gas emissions during operation. However, there can be some emissions from drilling and resource extraction, and potential issues such as induced seismicity and water usage must be managed.

 

5. Tidal Energy

Tidal energy exploits the movement of tides to generate electricity, with a carbon footprint of about 10-30 gCO₂e/kWh. Tidal energy systems produce very low emissions during operation. Nevertheless, the construction of tidal barrages or underwater turbines can impact marine ecosystems and alter tidal patterns.

 

Conclusion

While renewable energy sources are far cleaner than fossil fuels, they are not without environmental impact. The carbon footprints of solar, biomass, geothermal, and tidal energy sources highlight the importance of considering the full life-cycle emissions and other environmental impacts associated with each technology. As we continue to innovate and improve renewable energy technologies, reducing their carbon footprints and mitigating their environmental impacts will be crucial for achieving a truly sustainable energy future.

 

By understanding these impacts, we can make more informed decisions and support the development of cleaner and more efficient energy solutions. Let's continue to drive innovation and work towards a greener planet.

What is PLC? Progeammable Logic Controller.

 

  What is PLC?

Programmable Logic Controller

 

1. ·          Ladder Diagram (LD)
2. ·     Sequential Function Charts (SFC)
3. ·         Function Block Diagram (FBD)
4. ·         Structured Text (ST)
5. ·         Instruction List (IL)

 

1. Ladder Logic (LD)

Logic Gate with truth table

  LADDER LOGIC

 

 

"Ladder" diagrams

 

Ladder diagrams are specialized schematics commonly used to document industrial control logic systems. They are called "ladder" diagrams because they resemble a ladder, with two vertical rails (supply power) and as many "rungs" (horizontal lines) as there are control circuits to represent. If we wanted to draw a simple ladder diagram showing a lamp that is controlled by a hand switch, it would look like this:

 

 

The "L₁" and "L₂" designations refer to the two poles of a 120 VAC supply, unless otherwise noted. L₁ is the "hot" conductor, and L₂ is the grounded ("neutral") conductor. These designations have nothing to do with inductors, just to make things confusing. The actual transformer or generator supplying power to this circuit is omitted for simplicity. In reality, the circuit looks something like this:

 

Typically in industrial relay logic circuits, but not always, the operating voltage for the switch contacts and relay coils will be 120 volts AC. Lower voltage AC and even DC systems are sometimes built and documented according to "ladder" diagrams:

 

 

So long as the switch contacts and relay coils are all adequately rated, it really doesn't matter what level of voltage is chosen for the system to operate with.

 

Note the number "1" on the wire between the switch and the lamp. In the real world, that wire would be labeled with that number, using heat-shrink or adhesive tags, wherever it was convenient to identify. Wires leading to the switch would be labeled "L₁" and "1," respectively. Wires leading to the lamp would be labeled "1" and "L₂," respectively.

These wire numbers make assembly and maintenance very easy. Each conductor has its own unique wire number for the control system that it's used in. Wire numbers do not change at any junction or node, even if wire size, color, or length changes going into or out of a connection point. Of course, it is preferable to maintain consistent wire colors, but this is not always practical. What matters is that any one, electrically continuous point in a control circuit possesses the same wire number. Take this circuit section, for example, with wire #25 as a single, electrically continuous point threading to many different devices:

 

 

In ladder diagrams, the load device (lamp, relay coil, solenoid coil, etc.) is almost always drawn at the right-hand side of the rung. While it doesn't matter electrically where the relay coil is located within the rung, it does matter which end of the ladder's power supply is grounded, for reliable operation.

 

Take for instance this circuit:

 

Here, the lamp (load) is located on the right-hand side of the rung, and so is the ground connection for the power source. This is no accident or coincidence; rather, it is a purposeful element of good design practice. Suppose that wire #1 were to accidently come in contact with ground, the insulation of that wire having been rubbed off so that the bare conductor came in contact with grounded, metal conduit. Our circuit would now function like this:

With both sides of the lamp connected to ground, the lamp will be "shorted out" and unable to receive power to light up. If the switch were to close, there would be a short- circuit, immediately blowing the fuse.

 

However, consider what would happen to the circuit with the same fault (wire #1 coming in contact with ground), except this time we'll swap the positions of switch and fuse (L₂ is still grounded):

 

This time the accidental grounding of wire #1 will force power to the lamp while the switch will have no effect. It is much safer to have a system that blows a fuse in the event of a ground fault than to have a system that uncontrollably energizes lamps, relays, or solenoids in the event of the same fault. For this reason, the load(s) must always be located nearest the grounded power conductor in the ladder diagram.

 

·         REVIEW:

·         Ladder diagrams (sometimes called "ladder logic") are a type of electrical notation and symbology frequently used to illustrate how electromechanical switches and relays are interconnected.

·         The two vertical lines are called "rails" and attach to opposite poles of a power supply, usually 120 volts AC. L₁ designates the "hot" AC wire and L₂ the "neutral" (grounded) conductor.

·         Horizontal lines in a ladder diagram are called "rungs," each one representing a unique parallel circuit branch between the poles of the power supply.

·         Typically, wires in control systems are marked with numbers and/or letters for identification. The rule is, all permanently connected (electrically common) points must bear the same label.

 

 

Digital logic functions

 

We can construct simply logic functions for our hypothetical lamp circuit, using multiple contacts, and document these circuits quite easily and understandably with additional rungs to our original "ladder." If we use standard binary notation for the status of the switches and lamp (0 for un-actuated or de-energized; 1 for actuated or energized), a truth table can be made to show how the logic works:

 

Now, the lamp will come on if either contact A or contact B is actuated, because all it takes for the lamp to be energized is to have at least one path for current from wire L₁ to wire 1. What we have is a simple OR logic function, implemented with nothing more than contacts and a lamp.

 

We can mimic the AND logic function by wiring the two contacts in series instead of parallel:

 

 

Now, the lamp energizes only if contact A and contact B are simultaneously actuated. A path exists for current from wire L₁ to the lamp (wire 2) if and only if both switch contacts are closed.

 

The logical inversion, or NOT, function can be performed on a contact input simply by using a normally-closed contact instead of a normally-open contact:

 

 

Now, the lamp energizes if the contact is not actuated, and de-energizes when the contact

is actuated.

 

If we take our OR function and invert each "input" through the use of normally-closed contacts, we will end up with a NAND function. In a special branch of mathematics known as Boolean algebra, this effect of gate function identity changing with the inversion of input signals is described by DeMorgan's Theorem, a subject to be explored in more detail in a later chapter.

 

 

The lamp will be energized if either contact is un-actuated. It will go out only if both

contacts are actuated simultaneously.

 

Likewise, if we take our AND function and invert each "input" through the use of normally-closed contacts, we will end up with a NOR function:

 

 

A pattern quickly reveals itself when ladder circuits are compared with their logic gate counterparts:

 

·         Parallel contacts are equivalent to an OR gate.

·         Series contacts are equivalent to an AND gate.

·         Normally-closed contacts are equivalent to a NOT gate (inverter).

 

We can build combinational logic functions by grouping contacts in series-parallel arrangements, as well. In the following example, we have an Exclusive-OR function built from a combination of AND, OR, and inverter (NOT) gates:

 

 

The top rung (NC contact A in series with NO contact B) is the equivalent of the top NOT/AND gate combination. The bottom rung (NO contact A in series with NC contact

B)  is the equivalent of the bottom NOT/AND gate combination. The parallel connection between the two rungs at wire number 2 forms the equivalent of the OR gate, in allowing either rung 1 or rung 2 to energize the lamp.

 

To make the Exclusive-OR function, we had to use two contacts per input: one for direct input and the other for "inverted" input. The two "A" contacts are physically actuated by the same mechanism, as are the two "B" contacts. The common association between contacts is denoted by the label of the contact. There is no limit to how many contacts per switch can be represented in a ladder diagram, as each new contact on any switch or relay (either normally-open or normally-closed) used in the diagram is simply marked with the same label.

 

Sometimes, multiple contacts on a single switch (or relay) are designated by a compound labels, such as "A-1" and "A-2" instead of two "A" labels. This may be especially useful if you want to specifically designate which set of contacts on each switch or relay is being used for which part of a circuit. For simplicity's sake, I'll refrain from such elaborate labeling in this lesson. If you see a common label for multiple contacts, you know those contacts are all actuated by the same mechanism.

If we wish to invert the output of any switch-generated logic function, we must use a relay with a normally-closed contact. For instance, if we want to energize a load based on the inverse, or NOT, of a normally-open contact, we could do this:

 

 

We will call the relay, "control relay 1," or CR₁. When the coil of CR₁ (symbolized with the pair of parentheses on the first rung) is energized, the contact on the second rung opens, thus de-energizing the lamp. From switch A to the coil of CR₁, the logic function is non-inverted. The normally-closed contact actuated by relay coil CR₁ provides a logical inverter function to drive the lamp opposite that of the switch's actuation status.

 

Applying this inversion strategy to one of our inverted-input functions created earlier, such as the OR-to-NAND, we can invert the output with a relay to create a non-inverted function:

 

 

From the switches to the coil of CR₁, the logical function is that of a NAND gate. CR₁'s normally-closed contact provides one final inversion to turn the NAND function into an AND function.

 

·         REVIEW:

·         Parallel contacts are logically equivalent to an OR gate.

·         Series contacts are logically equivalent to an AND gate.

·         Normally closed (N.C.) contacts are logically equivalent to a NOT gate.

·         A relay must be used to invert the output of a logic gate function, while simple normally-closed switch contacts are sufficient to represent inverted gate inputs.