Under what circumstance does the integral $\int_{x_{0}}^{\infty} x^{b} d x$ diverge? Use this to argue that a physically acceptable wave function must fall to 0 faster than $|x|^{-\frac{1}{2}}$ does as x gets large.

Draw a 2-shell-passes and 8-tube-passes shell-and-tube heat exchanger. What is the primary reason for using so many tube passes?

Two flows are mixed to form a single flow. Flow at state 1 is 1.5 kg/s of water at 400 kPa, 200$^{\circ} \mathrm{C}$, and flow at state 2 is at 500 kPa, 100$^{\circ} \mathrm{C}$. Which mass flow rate at state 2 will produce an exit T3 = 150$^{\circ} \mathrm{C}$ if the exit pressure is kept at 300 kPa?

An electron is trapped in an infinite square-well potential of width 0.70 nm. If the electron is initially in the (n=4) state, what are the various photon energies that can be emitted as the electron jumps to the ground state?

Imagine two indistinguishable particles that share an attraction. All other things being equal, would you expect their multiparticle spatial state to be symmetric, antisymmetric, or neither? Explain.

Water is boiled at $100^\circ C$ by a spherical platinum heating element of diameter 15 cm and emissivity 0.10 immersed in the water. If the surface temperature of the heating element is $350^\circ C,$ determine the convective boiling heat transfer coefficient.

A 15 mm × 15 mm silicon chip is mounted such that the edges are flush in a substrate. The chip dissipates 1.4 W of power uniformly, while air at $20^\circ C$ (1 atm) with a velocity of 25 m/s is used to cool the upper surface of the chip. If the substrate provides an unheated starting length of 15 mm, determine the surface temperature at the trailing edge of the chip. Evaluate the air properties at $50^\circ C.$

Someone claims that the mass and the mole fractions for a mixture of $CO_2$ and $N_{2}O$ gases are identical. Do you agree? Explain.

A single pass heat exchanger is to be designed to heat 100,000 lbm of water in an hour from $60^\circ F$ to $100^\circ F$ by condensation of water vapor at $230^\circ F$ on the shell side. Each tube has an inner diameter of 1.2 in. and a wall thickness of 0.12 in. The inner surface convection heat transfer coefficient is $480 Btu/h \cdot ft^2 \cdot ^\circ F$ and the outer surface convection heat transfer coefficient is $2000 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot^{\circ} \mathrm{F}.$ If the inlet velocity of water $\left(c_{p c}=1 \mathrm{B} \mathrm{tu} / \mathrm{lbm} \cdot^{\circ} \mathrm{F} \text { and } \rho=62.3 \mathrm{lbm} / \mathrm{ft}^{3}\right)$ on the tube side is 4 ft/s, determine the required number of tubes and length of tubes. Assume the thermal resistance of the tubes is negligible.

What are the heat transfer mechanisms involved during heat transfer in a liquid-to-liquid heat exchanger from the hot to the cold fluid?

Consider an equimolar mixture of $CO_2$ and $O_2$ gases at 800 K and a total pressure of 0.5 atm. For a path length of 1.2 m, determine the emissivity of the gas.

Consider an ordinary house with R-13 walls (walls that have an R-value of $13 h \cdot ft^2 \cdot ^\circ F/Btu).$ Compare this to the R-value of the common double-door windows that are double pane $\frac{1}{4}$ with in of air space-and have aluminum frames. If the windows occupy only 20 percent of the wall area, determine if more heat is lost through the windows or through the remaining 80 percent of the wall area. Disregard infiltration losses.

A plate (0.5 m × 0.5 m) is inclined at an angle of $30^\circ.$ The top surface of the plate is well insulated. Estimate the rate of heat loss from the plate when the bottom surface is maintained at $60^\circ C$ and the surrounding atmospheric quiescent air is at $0^\circ C.$

Jakob suggests the following correlation be used for square tubes in a liquid cross-flow situation:$\mathrm{Nu}=0.102 \mathrm{Re}^{0.625} \mathrm{Pr}^{1 / 3}$ Water $(k=0.61 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}, \mathrm{Pr}=6)$ flows across a l-cm square tube with a Reynolds number of 10,000. The convection heat transfer coefficient is(a) $5.7 kW/m^2 \cdot K$ (b) $8.3 kW/m^2 \cdot K$ (c) $11.2 kW/m^2 \cdot K$ (d) $15.6 kW/m^2 \cdot K$ (e) $18.1 kW/m^2 \cdot K$