Bernoulli Equation Calculator
Compute bernoulli equation using validated scientific equations. See step-by-step derivations, unit analysis, and reference values.
Calculator
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Complete Solution
Energy Components (Pa)
Formula
Bernoulli's equation states that the total mechanical energy per unit volume is constant along a streamline in steady, incompressible, inviscid flow. The three terms represent static pressure energy, kinetic (dynamic) energy, and gravitational potential energy. Given five of the six variables (P, v, h at two points), the sixth can be calculated.
Last reviewed: December 2025
Worked Examples
Example 1: Venturi Flow Meter
Example 2: Water Tank Drainage (Torricelli)
Background & Theory
The Bernoulli Equation Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Bernoulli Equation Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Formula
P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂
Bernoulli's equation states that the total mechanical energy per unit volume is constant along a streamline in steady, incompressible, inviscid flow. The three terms represent static pressure energy, kinetic (dynamic) energy, and gravitational potential energy. Given five of the six variables (P, v, h at two points), the sixth can be calculated.
Worked Examples
Example 1: Venturi Flow Meter
Problem: Water (ρ = 998 kg/m³) flows at 2 m/s in a 100mm pipe that narrows to 50mm. If inlet pressure is 200 kPa, find the pressure at the throat.
Solution: By continuity: A₁v₁ = A₂v₂\nv₂ = v₁ × (D₁/D₂)² = 2 × (100/50)² = 8 m/s\nBernoulli: P₂ = P₁ + ½ρ(v₁² - v₂²)\nP₂ = 200000 + 0.5 × 998 × (4 - 64)\nP₂ = 200000 - 29940 = 170,060 Pa = 170.1 kPa\nPressure drop = 29.9 kPa
Result: P₂ = 170.1 kPa | ΔP = 29.9 kPa | v₂ = 8 m/s
Example 2: Water Tank Drainage (Torricelli)
Problem: A water tank has an opening 3m below the water surface. Both the surface and opening are at atmospheric pressure. Find the exit velocity.
Solution: P₁ = P₂ = atmospheric (cancel out)\nv₁ ≈ 0 (large tank surface)\nh₁ = 3m, h₂ = 0m\nBernoulli: ρgh₁ = ½ρv₂²\nv₂ = √(2gh) = √(2 × 9.81 × 3) = 7.67 m/s\nThis is Torricelli's theorem
Result: v₂ = 7.67 m/s | Equivalent to free fall from 3m
Frequently Asked Questions
What is Bernoulli's equation?
Bernoulli's equation is a fundamental principle in fluid dynamics that describes the conservation of energy along a streamline in a steady, incompressible, inviscid flow. It states: P + ½ρv² + ρgh = constant, where P is static pressure (Pa), ρ is fluid density (kg/m³), v is flow velocity (m/s), g is gravitational acceleration (9.81 m/s²), and h is elevation (m). The three terms represent pressure energy, kinetic energy, and potential energy per unit volume. The equation shows that as fluid velocity increases, pressure decreases (and vice versa), which explains airplane lift, Venturi effect, and many other phenomena.
What are the assumptions of Bernoulli's equation?
Bernoulli's equation requires several assumptions: (1) Steady flow — flow properties don't change with time at any point. (2) Incompressible fluid — density remains constant (valid for liquids and low-speed gases below Mach 0.3). (3) Inviscid flow — no friction losses (real flows have losses, accounted for by adding a head loss term). (4) Flow along a streamline — the equation connects two points on the same streamline. (5) No energy addition or removal — no pumps, fans, or turbines between the points. When these assumptions don't hold, modified forms (with loss terms, compressibility corrections, or shaft work) are used.
How does Bernoulli's equation explain the Venturi effect?
The Venturi effect occurs when fluid flows through a constriction (narrowing). By the continuity equation (A₁v₁ = A₂v₂), velocity must increase in the narrow section. Bernoulli's equation then dictates that pressure must decrease to conserve energy: P₁ + ½ρv₁² = P₂ + ½ρv₂². Since v₂ > v₁, then P₂ < P₁. This pressure drop is used in: Venturi flow meters, carburetors (drawing fuel into airflow), aspirators and ejectors, atomizers and spray nozzles. The Venturi effect is essentially the conversion of pressure energy to kinetic energy.
How does Bernoulli's equation relate to Torricelli's theorem?
Torricelli's theorem is a special case of Bernoulli's equation for liquid draining from a tank through an orifice. Applying Bernoulli between the liquid surface (v₁ ≈ 0, P₁ = P_atm) and the orifice (P₂ = P_atm): ρgh = ½ρv₂², giving v₂ = √(2gh). The exit velocity equals the velocity of an object in free fall from height h. For example, water in a tank with the hole 2m below the surface: v = √(2 × 9.81 × 2) = 6.26 m/s. This assumes an ideal fluid; real discharge is modified by a coefficient of discharge (typically 0.6-0.65 for a sharp-edged orifice).
How do I get the most accurate result?
Enter values as precisely as possible using the correct units for each field. Check that you have selected the right unit (e.g. kilograms vs pounds, meters vs feet) before calculating. Rounding inputs early can reduce output precision.
How do I verify Bernoulli Equation Calculator's result independently?
The Formula section on this page shows the equation used. You can reproduce the calculation manually or in a spreadsheet using those steps. Compare your answer against the worked examples in the Examples section, which use known reference values so you can confirm the calculator is behaving as expected.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy