\documentclass[fontsize=12pt, paper=a4]{article}
\usepackage{xcolor}
\usepackage{graphicx}
\usepackage[a4paper, total={7in, 9in}]{geometry}
\usepackage{mathrsfs}
\title{Problems of Chapter 1}
%\author{Yi-Chao XIE}
\begin{document}
\maketitle

\textcolor{red}{Relation between the units:
\begin{center}
 1 ft=0.3048m;1lb=0.454 kg; 1lb/ft$^2$=47.89N/m$^2$=47.89 Pa; $1^o$R=5/9K\\
 \end{center}}
 
 \textcolor{red}{
\begin{center}
 \bf{Read carefully chapter 1, especially Ch 1.3 and 1.4. You will found the questions are rather easy to solve :). \\Deadline: September 22nd, 2023 }
  \end{center}}
 
 
\vspace{1cm}
1.1 At the nose of a missile in flight, the pressure and temperature are 5.6 atm
and 850°R, respectively. Calculate the density and specific volume.
(Note: 1 atm $=$ 2116 lb/ft$^2$.)


\vspace{7cm}
1.2 In the reservoir of a supersonic wind tunnel, the pressure and temperature of air are 10 atm and 320 K, respectively. Calculate the density, the number density, and the mole-mass ratio. (Note: 1 atm = $1.01 \times 10^5$ N/m$^2$.)


\vspace{7cm}
1.3 For a calorically perfect gas, derive the relation $c_p-c_v=R$. Repeat the
derivation for a thermally perfect gas.




\vspace{7cm}
1.4 The pressure and temperature ratios across a given portion of a shock wave in air are $p_2/p_1=$ 4.5 and $T_2/T_1=$ 1.687, where 1 and 2 denote conditions ahead of and behind the shock wave, respectively. Calculate the change in entropy in units of (a) (ft $\cdot$ lb)/(slug$\cdot$$^o$R) and (b) J/(kg$\cdot$ K).



\vspace{7cm}
1.5 Assume that the flow of air through a given duct is isentropic. At one point in the duct, the pressure and temperature are $p_1$ = 1800 lb/ft$^2$ and $T_1$ = 500 $^o$R, respectively. At a second point, the temperature is 400 $^o$R. Calculate the pressure and density at this second point.




\vspace{7cm}
1.6 Consider a room that is 20 ft long, 15 ft wide, and 8 ft high. For standard sea level conditions, calculate the mass of air in the room in slugs. Calculate the weight in pounds. (Note: If you do not know what standard sea level conditions are, consult any aerodynamics text, such as Refs. 1 and 104, for these values. Also, they can be obtained from any standard atmosphere table.)



\vspace{7cm}
1.7 In the infinitesimal neighborhood surrounding a point in an inviscid flow, the small change in pressure, $dp$, that corresponds to a small change in velocity, $dV$, is given by the differential relation $dp=-\rho VdV$. (This equation is called Euler’s Equation; it is derived in Chap. 6.)
a. Using this relation, derive a differential relation for the fractional change in density, $d\rho/\rho$, as a function of the fractional change in velocity, $dV/V$, with the compressibility $\tau$ as a coefficient.\\
b. The velocity at a point in an isentropic flow of air is 10 m/s (a low speed flow), and the density and pressure are 1.23 kg/m$^3$ and $1.01\times 10^5$ N/m$^2$, respectively (corresponding to standard sea level conditions). The fractional change in velocity at the point is 0.01. Calculate the fractional change in density. \\
c. Repeat part (b), except for a local velocity at the point of 1000 m/s (a high-speed flow). Compare this result with that from part (b), and comment on the differences.



\end{document}