Case studies and course project

This course has built the calculus of several variables step by step: functions and their graphs, curves and motion, partial derivatives and optimization, multiple integrals and changes of variables, line and surface integrals, the theorems of Green, Gauss and Stokes, and their use in fluid mechanics and electromagnetism. Real engineering questions rarely announce which of these tools they need. This lesson works through three case studies that each combine several lessons: the wind force and overturning moment on a tall building, the best tilt for a solar panel, and laminar flow of oil in a pipe. It then summarizes the common approach, sets out the course project and looks ahead to the courses that rely on this one.

Case study 1: wind load on a tall building

Wind speed increases with height above the ground because the ground slows the air near it. A common model is the power law \(v(z) = v_{10}(z/10)^{\alpha}\), where \(v_{10}\) is the speed at 10 m and the exponent α depends on the terrain (about 0.15 over open water, 0.2 to 0.3 over suburbs and cities). The pressure on the windward face is \(p = C_p\,\tfrac{1}{2}\rho v^2\), with a pressure coefficient \(C_p\) of about 0.8. The total force and the overturning moment about the base are integrals over the face (Lesson 5), and the height of the resultant follows from the ratio, exactly as for the center of pressure on a gate (Lesson 6).

import numpy as np

rho, v10, alpha, Cp = 1.2, 30.0, 0.2, 0.8
B, H = 30.0, 60.0
z = np.linspace(0, H, 6001)
q = 0.5 * rho * (v10 * (z / 10)**alpha)**2          # dynamic pressure, Pa

F = Cp * B * np.trapezoid(q, z)
M = Cp * B * np.trapezoid(q * z, z)
k = 0.5 * rho * v10**2 * 10**(-2 * alpha)
F_exact = Cp * B * k * H**(1 + 2 * alpha) / (1 + 2 * alpha)
print(f"Numerical:   F = {F / 1e6:.3f} MN, M = {M / 1e6:.1f} MN m, height {M / F:.1f} m")
print(f"Closed form: F = {F_exact / 1e6:.3f} MN")
print(f"Uniform top pressure:   F = {Cp * B * H * q[-1] / 1e6:.3f} MN")
print(f"Uniform 10 m pressure:  F = {Cp * B * H * 0.5 * rho * v10**2 / 1e6:.3f} MN")

Numerical:   F = 1.137 MN, M = 39.8 MN m, height 35.0 m
Closed form: F = 1.137 MN
Uniform top pressure:   F = 1.592 MN
Uniform 10 m pressure:  F = 0.778 MN

Assuming the top-of-building pressure over the whole face overestimates the force by about 40%; using the 10 m pressure everywhere underestimates it badly. Integrating the actual profile is both more accurate and more economical.

Case study 2: the best tilt for a solar panel

The power a solar panel receives is the flux of sunlight through it (Lesson 7): if the sunlight has intensity \(I\) in the direction of the unit vector \(\mathbf{s}\) toward the sun, a panel of area A with unit normal \(\mathbf{n}\) receives \(IA\,\mathbf{s} \cdot \mathbf{n} = IA\cos\theta\), where θ is the angle of incidence, and nothing when the sun is behind it. The sun's direction changes through the day and the year, so the energy collected is an integral over time, and the best fixed tilt is an optimization problem (Lesson 4).

Set up axes with x east, y north and z up, at latitude φ. On the day of an equinox, when the sun rises due east and sets due west, the sun direction at hour angle h (15° per hour from solar noon) is \(\mathbf{s} = (-\sin h, -\sin\varphi\cos h, \cos\varphi\cos h)\) (h is negative in the morning, when the sun is in the east). A panel facing south and tilted at angle β from the horizontal has normal \(\mathbf{n} = (0, -\sin\beta, \cos\beta)\). Their dot product simplifies, with a trigonometric identity, to

Case study 3: laminar flow in a pipe

For slow, steady flow of a viscous fluid along a straight pipe of radius R, the velocity u is along the axis and depends only on the radius r. The Navier-Stokes equations (Lesson 10) reduce to a balance between the pressure gradient G = −dp/dx (positive when pressure falls along the pipe) and viscous friction, written with the cylindrical Laplacian (Lesson 9):

Integrating twice, with u finite on the axis and zero at the wall (no slip), gives the parabolic Poiseuille profile:

The flow rate is the flux of velocity through the cross-section, a double integral in polar coordinates (Lesson 5): \(Q = \int_0^{2\pi}\int_0^R u\,r\,dr\,d\theta = \dfrac{\pi GR^4}{8\mu}\). The mean velocity is half the maximum, and the wall shear stress is \(\tau_w = \mu|du/dr|_{r=R} = GR/2\).

Common threads

The three studies used different tools but the same approach:

• Describe the geometry with suitable coordinates: heights on a facade, vectors toward the sun, radius in a pipe.
• Express the physics as fields (pressure, light flux, velocity) and the quantities wanted as integrals, derivatives or extremes of those fields.
• Exploit symmetry to reduce dimensions: the pipe problem collapsed from three dimensions to one.
• Solve in closed form where possible, and check numerically, or the reverse.
• Check limits and validity: the Reynolds number for laminar flow, the uniform-pressure estimate for wind, the atmosphere neglected for sunlight.
• Interpret: a resultant height for the foundation designer, a tilt for the installer, a flow rate for the lubrication system.

The course project

Choose one of the following problems, or propose your own of similar scope with your instructor's approval, and prepare a technical report with your formulation, analysis, results (analytical and numerical) and interpretation.

• Project A: Terrain and earthworks. Obtain or construct a contour map of a site. Model the surface as a function of two variables, find slopes and drainage directions from the gradient, locate high and low points, and compute the cut and fill volumes needed to level a building pad by double integration.
• Project B: Optimal design. Choose a component (a tank, beam, duct, heat sink or panel layout) with at least two design variables and one constraint. Formulate the objective, solve with Lagrange multipliers and numerically, interpret the multiplier, and examine how the optimum changes if the objective or constraint changes.
• Project C: Flow field. Construct a two-dimensional potential flow from sources, sinks, vortices and uniform flow to model a practical situation (flow past a bridge pier, toward a well field or around a building). Plot streamlines, compute pressures with Bernoulli's equation, verify incompressibility and irrotationality, and discuss where the ideal-flow model fails.
• Project D: Field computation. Solve Laplace's equation numerically for an electrostatic, heat conduction or seepage problem in a non-rectangular region. Check your solver against an exact solution (a coaxial cable or a pipe wall), then compute a quantity of engineering interest such as capacitance, heat loss or seepage flow.

Every report should state the physical laws and assumptions, show the coordinate system and any changes of variables, verify results by an independent method, discuss sensitivity to uncertain inputs and conclude with clear engineering recommendations.

Working with software

Engineering software for structures, fluids, heat and electromagnetics is built on the mathematics of this course: finite element and finite volume programs integrate over elements, apply the divergence theorem to each cell and solve the resulting equations; optimization tools compute gradients and Hessians; geographic and surveying software works with surfaces and contours. Using such tools well requires knowing what they compute, choosing meshes and tolerances sensibly, and checking results against hand calculations like those in this lesson.

Looking ahead

Multivariable calculus is the language of the engineering sciences. Fluid mechanics, heat transfer, electromagnetics, solid mechanics and geotechnics all express their laws with gradients, divergences, curls and integrals over regions and surfaces. Differential equations (MATH-201) solve the resulting equations in simple cases, and numerical methods (MATH-300) solve them in general. Statistics (STAT-235) uses multiple integrals for joint distributions. The geometric intuition built here, of fields, flux and circulation, will make each of those subjects easier to understand.