Chapter 7, Problem 7.13P

Note: In the following problems, you will deal with both the International System of Units (SI) (N, kg, m, s, K) and the English Engineering System (lb, slug, ft, s, $°\text{R}$). Which system to use will be self-evident in each problem. All problems deal with calorically perfect air as the gas, unless otherwise noted. Also, recall that $1\text{atm}=2116{\text{lb/ft}}^{\text{2}}=1.01\times {10}^5{\text{N/m}}^{\text{2}}$.

Bernoulli’s equation, Equation $\left(3.13\right),\left(3.14\right)$, or $\left(3.15\right)$, was derived in Chapter 3 from Newton’s second law; it is fundamentally a statement that force $=$ mass $\times$ acceleration. However, the terms in Bernoulli’s equation have dimensions of energy per unit volume (check it out), which prompt some argument that Bernoulli’s equation is an energy equation for incompressible flow. If this is so, then it should be derivable from the energy equation for compressible flow discussed in the present chapter. Starting with Equation $\left(7.53\right)$ for inviscid, adiabatic compressible flow, make the appropriate assumptions for an incompressible flow and see what you need to do to obtain Bernoulli’s equation.