Chapter 3.9, Problem 17P

(A)

Interpretation Introduction

Interpretation:

The reasonable estimate of the velocity $\left(v\right)$ of a raindrop.

The kinetic energy $\left(\text{K}\text{.E}\right)$ of the rain.

Concept Introduction:

Write the expression to calculate the kinetic energy $\left(\text{K}\text{.E}\right)$.

$\text{K}\text{.E}=\frac{1}{2}m{v}^2$

Here, mass of the rain fall is $m$.

(B)

Interpretation Introduction

Interpretation:

Better estimation to note the values from the steam tables.

(C)

Interpretation Introduction

Interpretation:

The net amount of heat $\left(Q_{net}\right)$ that was added to or removed from the lake

Concept Introduction:

The energy balance equation for lake.

$\Delta \left\{M\left(\widehat{U}+\frac{V^2}{2}+gh\right)\right\}=\left[\begin{array}{l}{\displaystyle \sum _{j=1}^{j=J}\left\{m_{j,in}\left({\widehat{H}}_j+\frac{V_j^2}{2}+g{h}_j\right)\right\}}\\ -{\displaystyle \sum _{k=1}^{k=K}\left\{m_{k,out}\left({\widehat{H}}_k+\frac{V_k^2}{2}+g{h}_k\right)\right\}}+W_{EC}+W_S+Q\end{array}\right]$

Here, mass of the system is $M$, specific internal energy is $\widehat{U}$, velocity is $V$, acceleration due to gravity is $g$, height is $h$, individual quantities of mass added to and removed from the process are $m_{j,in}\text{and}{m}_{k,out}$ respectively, initial specific enthalpy is ${\widehat{H}}_j$, final specific enthalpy is ${\widehat{H}}_k$, initial velocity is $V_j$, final velocity is $V_k$, initial height is $h_j$, final height is $h_k$, shaft work addition is $W_{\text{S}}$, work addition through expansion or contraction of the system is $W_{\text{EC}}$, and heat addition is $Q$.

Rewrite the steady state equation by neglecting the work and mass flow rate.

$0=m_{in}{\widehat{H}}_{in}-m_{out}{\widehat{H}}_{out}+Q_{net}$

(D)

Interpretation Introduction

Interpretation:

The net heat transfer.

Concept Introduction:

Rewrite the steady state Equation (1).

$\begin{array}{l}M\left({\widehat{U}}_{final}-{\widehat{U}}_{initial}\right)=m_{in}{\widehat{H}}_{in}-m_{out}{\widehat{H}}_{out}+Q_{net}\\ M{C}_P\Delta T=m_{in}{\widehat{H}}_{in}-m_{out}{\widehat{H}}_{out}+Q_{net}\\ M{C}_P\left(T_{final}-T_{initial}\right)=m_{in}{\widehat{H}}_{in}-m_{out}{\widehat{H}}_{out}+Q_{net}\end{array}$

Here, mass of the lake is $M$, initial specific internal energy is ${\widehat{U}}_{initial}$, and final specific internal energy is ${\widehat{U}}_{final}$, constant pressure heat capacity is $C_P$, final temperature is $T_{final}$, and initial temperature is $T_{initial}$.

(E)

Interpretation Introduction

Interpretation:

The net heat transfer

Concept Introduction:

The net heat transfer using steady state equation.

$M_{final}{\widehat{U}}_{final}-M_{initial}{\widehat{U}}_{initial}=m_{in}{\widehat{H}}_{in}-m_{out}{\widehat{H}}_{out}+Q_{net}$

Here, final mass is $M_{final}$ and initial mass is $M_{initial}$.

The mass flow rate equation.

$\begin{array}{l}{M}_{final}-M_{initial}=m_{in}-m_{out}\\ M_{final}=m_{in}-m_{out}+M_{initial}\end{array}$