Difference Form of Entropy Balance:

General Equation: $S_{t_2}-S_{t_1}={\displaystyle \underset{k=1}{\overset{K}{}}}({\displaystyle \underset{t_1}{\overset{t_2}{}}}{M}_k^{}$hat$(S{)}_{k}dt)+{\displaystyle \underset{t_1}{\overset{t_2}{}}}\frac{(Q^{})}{T}dt+S_{\text{g}\text{e}\text{n}}$

Problem $1$:

The a cylinder is attached to a rectangular prism as shown in the following figure. The diameter

of the cylinder is $0\mathrm{.}5m,$ and the height of the cylinder is $3m.$ The length of the prism is $3m,$ while

the height and width are $0\mathrm{.}8m$ and $0\mathrm{.}9m,$ respectively. The thickness of the cylinder bottom is

$0\mathrm{.}015m,$ and the thickness of prism wall is $0\mathrm{.}01m.$ The outer surface of the prism and cylinder is

well$-$insulated except for the connected area.

At a specific time $t,$ the prism is filled with liquid A with a uniform temperature of $360K,$ and the

cylinder is filled with liquid B with a uniform temperature of $390K.$ This leads to heat transfer

between the prism and the cylinder. The thermal conductivity of the connected area is $135\frac{W}{K}\frac{?}{m}.$

Assume this heat transfer happens slowly enough that either liquid is in approximate equilibrium

at any given instant. $($i$.$e$.,$ there is little spatial variation in temperature within each shape, and we

can say $S_{\text{g}\text{e}\text{n}}^{}$~~$0$ when considering either liquid separately. This is called a quasi$-$static process.$)$

For that instant in time:

a$)$ Calculate the heat transfer rate for two subsystems $($prism and cylinder$)$

b$)$ Calculate the entropy change for the prism subsystem

c$)$ Calculate the entropy change for the cylinder subsystem

d$)$ Calculate the entropy change of the whole system $($view the combined prism and

cylinder as the system$)$

Differential Form of Entropy Balance:

neral Form: $\frac{dS}{dt}={\displaystyle \underset{k=1}{\overset{K}{}}}(M_k^{}$hat$(S{)}_k)+\frac{(Q^{})}{T}+S_{\text{g}\text{e}\text{n}}^{}$

This $is$ a thermodynamics question. Answers

$1$a$)$

$1$b$)88\mathrm{.}4$

$1$c$)-81\mathrm{.}6$

$1$d$)6\mathrm{.}80$