$4$As can be seen above, the detector cell can be modeled to have annular region for gas flow.

The interior surface is at Tf$,$ the filament temperature. The outer surface is at Tw$,$ the

temperature of the cell walls.

a$.$ Show that the steady$-$state temperature profile between the filament and the cell

walls can be derived as:

$()$

$()$

$()$

$1$

$21$

ln

ln f f w

r R

T T T T

R R

b$.$ The cell wall temperature, Tw depends on the rate of heat removal througgh a layer of insulation that surrounds the cell. Nothe that at steady state, a constant amount of heat is generated at the filament, the majority of which must be carried radially through the gas and must be expelled from the cell walls. The heat loss from the cell walls can be expressed as $Q^{},$ whereas the heat generated at the filament is $I^2{R}_{},$ where $R_e$ is the electrical resistance of the filament, which is dependent of filament temperature. The heat loss is through a layered structure and the overall heat transfer coefficient for heat transfer between the cell wall and the ambient is $U.$ Using this information, provide an expression for $T_w.$

c$.$ The thermal conductivities of $H_2$ and $He$ are the greatest among gases $.$ Whenever a different gas entrained in this these carrier gases pass through the cell, the

effective $k$ will be smaller, and this will cause a drop in the rate of heat removal from the filament. Most manufacturers design TCDs in a way that the filament is connected to a circuit that keeps $T_f$ constant by changing the current. The change in the current supplied is measured, which constitutes the signal extracted from the detector. By using the expressions you derived in parts a & b$,$ show that I can be explicitly related to operation, geometrical and physical parameters as:

$I=\{\frac{2kL}{R_e}[\frac{T_f-T_{}}{ln(\frac{R_2}{R_1})+\frac{\frac{k}{R_2}}{U}}]{\}}^{\frac{1}{2}}$

This is the design equation for the TCD$.$ For a given set of Tf $($operation

temperature$),$ geometrical parameters R$1,$ R$2$ and L$,$ and the structure of the outer

insulation $($its information is buried under U$),$ the sensitivity of I with respect to k

can be obtained.