Bernoulli's Principle and Venturi Meters

• Bernoulli's principle states that a fluid's speed increase coincides with a pressure decrease or lower potential energy.

Bernoulli's Principle Equation

• P + 1/2 ρ v^2 = constant
  • P is static pressure
  • ρ is fluid density
  • v is fluid velocity

Venturi Meter

• The Venturi effect is a fluid pressure reduction when it flows through a constricted pipe section.
• Fluid flow rate is measured by measuring pressure at different pipe sections.
• A1v1 = A2v2 (Continuity equation)
  • A1 is area at point 1
  • A2 is area at point 2
  • v1 is velocity at point 1
  • v2 is velocity at point 2
• P1 + 1/2 ρ v1^2 = P2 + 1/2 ρ v2^2 (Bernoulli's equation)
  • P1 is the pressure at point 1
  • P2 is the pressure at point 2

Fonction Logarithme Népérien (Natural Logarithm Function)

• The natural logarithm function, denoted ln, is defined on ]0; +∞[ and is the primitive of the function $x \mapsto \frac{1}{x}$ that vanishes at 1.

Propriétés (Properties)

• ln(1) = 0
• ln(e) = 1
• For all x > 0, $\ln(x) = \int_1^x \frac{1}{t} dt$
• The ln function is differentiable on ]0; +∞[ and for all x > 0, $\ln'(x) = \frac{1}{x}$
• The ln function is strictly increasing on ]0; +∞[

Propriétés Algébriques (Algebraic Properties)

• For all real numbers a > 0 and b > 0, and for all relative integers n:
  • ln(ab) = ln(a) + ln(b)
  • $\ln(\frac{a}{b}) = \ln(a) - \ln(b)$
  • $\ln(\frac{1}{b}) = -\ln(b)$
  • ln(an) = n ln(a)
  • $\ln(\sqrt{a}) = \frac{1}{2} \ln(a)$

Limites (Limits)

• $\lim_{x \to +\infty} \ln(x) = +\infty$
• $\lim_{x \to 0} \ln(x) = -\infty$
• $\lim_{x \to +\infty} \frac{\ln(x)}{x} = 0$
• $\lim_{x \to 0} x \ln(x) = 0$
• $\lim_{x \to 1} \frac{\ln(x)}{x-1} = 1$
• $\lim_{h \to 0} \frac{\ln(1+h)}{h} = 1$

Dérivées (Derivatives)

• Let u be a function differentiable and strictly positive on an interval I. Then the function ln(u) is differentiable on I and:
  • $(\ln(u))' = \frac{u'}{u}$
• Example:
  • If $f(x) = \ln(2x^2 + 1)$ then $f'(x) = \frac{4x}{2x^2 + 1}$

Courbe représentative (Representative Curve)

• The ln function is concave.

Exercices(Exercises)

• Solve the following equations:
  • ln(x) = 0
  • ln(x) = 1
  • ln(x) = -1
• Solve the following inequalities:
  • ln(x) > 0
  • ln(x) < 0
  • ln(x) ≥ 1
• Calculate the derivatives of the following functions:
  • $f(x) = \ln(x^2 + 1)$
  • $f(x) = \ln(2x + 3)$
  • $f(x) = x \ln(x)$