$r_1=$

We consider a cylindrical pipe described in the figure above. This pipe is made of a material having a thermal conductivity $k.$ Inner and outer radiuses are $r_i$ and $r_o=r_i+e_1$ respectively, where $e_1$ is the wall thickness of the pipe. The ambient external temperature is $T_o=20C.h_i$ and $h_0$ are the two coefficients of convective transfer inside and outside the pipe.

Let $T_1$ and $T_2$ be the temperatures of respectively the inner and outer surfaces of the pipe wall. Considering steady$-$state and only radial conduction, give the analytical equation governing the temperature profile across the pipe wall, as a function of $T_1,T_2,r_i$ and $r_o.$

Considering a motionless fluid in the pipe which has a constant temperature $T_f,$ calculate $R_{th}^{\text{'}}($in K$.$m$.W^{-1})$ the overall thermal resistance between the fluid and the ambient air per pipe's unit of length.

Given $T_f=150c,$ calculate $Q_l^{\text{'}}($in $\frac{W}{m})$ the heat rate lost through the pipe wall per unit of length. Fluid

The fluid is now moving with a mass flow rate $m_f$ and its temperature is not constant throughout the length of the pipe anymore. Write the steady$-$state energy balance of a control volume of fluid of length $dx.$ tempenture change through the fluid hove inthe

Give the expression of the fluid temperature $T(x)-T_{in}$ at any position x in the pipe.

Calculate the fluid temperature $T_{\text{o}\text{u}\text{t}}$ at the outlet of the pipe given a fluid mass flow rate $m_f=3k\frac{g}{s},$ a thermal capacity of the fluid $c_f=4\mathrm{.}2kJ*k{g}^{-1}*K^{-1}$ and an inlet temperature $T_{in}=150C.$

Data: $r_i=20mm$;$e_1=2mm$;$k=380W*m^{-1}*K^{-1}$;$L=100m$;$h_o=20W*m^{-2}*K^{-1}$; $h_i=200W*m^{-2}*K^{-1}$;