$($a$)$ Dry saturated steam at a temperature of $180$C is to be produced in a fire tube boiler from the cooling of $50000$ kg h$1$ of flue gases from a pressurised combustion process. The gases enter the tubes of the boiler at $1600$C and leave at $200$C$.$ The feed water is externally preheated to $180$C before entering the boiler.

The mean specific heat capacity of the flue gases is $1\mathrm{.}15$ kJ kg$1$ K$1.$ The latent heat of vaporisation of the water at $180$C is $2015$ kJ kg$1.$ Feed water temperature $=180$C$.$

Determine the amount of steam produced per hour, if the total heat loss is $10\%$ of the heat available for steam raising.

$($b$)$ The overall heat transfer coefficient based on the outside area of the tubes is given as $54$ W m$2$ K$1.$ Determine the area of heat transfer required to perform this duty.

$($c$)$ The tubes within the boiler are to be $25$ mm inside diameter with a wall thickness of $3$ mm$.$ The average flue gas velocity through the tubes to maintain the overall heat transfer coefficient value and to minimise pressure losses is to be more than $22$ m s$1$ and less than $28$ m s$1.$

Assuming that the average density of the flue gases is $1\mathrm{.}108$ kg m$3,$ calculate:

$($i$)$ the minimum and maximum number of tubes in each pass

$($ii$)$ the overall length of tubes at each of these numbers of tubes

$($iii$)$ the minimum number of tube passes in each case, if the lengthof a boiler tube is to be less than $5$ metres.