Problem $1.$ We wish to estimate the dynamics of a cold thermocouple probe suddenly placed in

a hot flowing fluid stream for the purpose of temperature measurement. The probe consists of

two dissimilar metal wires joined by soldering at the tip, and the wires are then encased in a

metal sheath and the tip is finally coated with a bead of plastic to protect it from corrosion. Take

the mass of the soldered tip plus plastic bead to be m$,$ with a specific heat C p $.$ Denote the transfer

coefficient as h$.$

a$.$ If the effects of thermal conductivity can be ignored, show that the temperature response

of the probe is described by:

where A denotes the exposed area of probe tip, and T$($t$)$ is its temperature.

b$.$ Lump the explicit parameters to form the system time constant, and define a new variable

T $*=($T f $$T$)$ and show that the compact form results

To do this you will need to show that the system time constant is defined as:

c$.$ Integrate the expression in $($b$),$ using the initial condition T$(0)=$ To and show that:

d$.$ Rearrange the expression in $($c$)$ to obtain the solution below.

mC p

dT

dt $=$ hA$($T f $$ T $)$

$\backslash$tau dT $*$

dt $=$T $*$

$\backslash$tau $=$ mC p

hA

T $$ T f

T$0$ T f

$=$ exp $$ t

$\backslash$tau

$$

$$

$$

T$0$ T

T$0$ T f

$=1$ exp $$ t

$\backslash$tau

$$

$$

$$

e$.$ What does the solution look like? Plot this up in Maple to find out. Show that the

temperature excess is $63\%$ of the final steady$-$state after a time equivalent of one time

constant has elapsed.

Problem $2.$ An iron bar $2$ cm x $3$ cm x $10$ cm at a temperature of $95$ o C is dropped into a barrel of

water at $25$ o C$.$ The barrel is large enough that the water temperature rises negligibly as the bar

cools. The rate at which heat is transferred from the bar to the water is given by

Q$($J$/$min$)=$ hA$($T b $-$T w $)$

Where U $(=0\mathrm{.}050$ J$/$min$-$cm$2$ o C$)$ is a heat transfer coefficient, A $($cm$2)$ is the exposed area of the

bar, T b is the surface temperature of the bar, and T w is the water temperature. Given that the

heat capacity of the bar is $0\mathrm{.}460$ J$/$g o C$,$ and heat conduction in iron is fast enough to assume that

the temperature T b $($t$)$ is uniform throughout the bar, write an energy balance on the bar and

determine the steady$-$state temperature of the bar. Also, calculate the time required for the bar

to cool to $30$ o C$.$

Problem $3.$ A steel ball initially at a uniform temperature of $100$o C is dropped into an insulated

vessel containing water at $20$o C$.$ Derive the appropriate governing equations for the temperature

of the steel ball $($T b $)$ and the temperature of the water $($T w $).$ Determine the transient and steady

state solutions for the temperature of water and steel ball as a function of the system

parameters. Plot these profiles for the set of parameters with the mass of the ball $($ mb $=1\mathrm{.}25$ kg$),$

the mass of the water $($mw $=5$ kg$),$ and the specific heats $($ Cb $=3360$ J$/$go C$,$ C w $=4200$ J$/$go C$)$ and hA

$($the heat transfer rate$)=4200$ J$/$s o C$.$ What do you observe?

Problem $4.$ An aluminum plate is at a temperature of $25$o C$.$ The plate is then suddenly subjected

to a uniform and continuing heating at the rate of $6$ W$/$m $2.$ Develop a macroscopic model for the

process. Find the time at which the surface temperature reaches a value of $100$o C$.$ What is the

total heat transferred up to this time? Assume a plate thickness of $10$ cm and that the heat

conduction is fast enough to assume the temperature is uniform throughout the plate.

Problem $5.$ Derive the governing differential equation for the concentration in a plug$-$flow $($i$.$e$.$

constant velocity$)$ reactor if the reaction rate is of a$)$ second order and b$)$ zeroth order. Render

each of these equations dimensionless and solve them numerically in Maple. Plot each solution

as a function of the dimensionless plug$-$flow reactor axial distance