Problems $1$: $(40$ points$)$

Atmospheric air enters at $\backslash (1\backslash$mathrm$\{$~atm$\},32^\{\backslash$circ$\}\backslash$mathrm$\{$C$\},95\backslash \%\backslash )$ relative humidity and volume flow rate of $\backslash (1000\backslash$mathrm$\{$~m$\}^\{3\}/\backslash$mathrm$\{$s$\}\backslash ).$ It is cooled to $\backslash (24^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ and $\backslash (60\backslash \%\backslash )$ relative humidity. A vapor$-$compression refrigeration system using R$-410$A as the working fluid is used to provide the cooling required. The evaporator operates at $\backslash (4^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ and the condenser at a saturation temperature of $\backslash (39\mathrm{.}4^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ The condenser rejects its heat to the atmospheric air.

Find:

a$.$ Draw the process in a $\backslash ($ T$-$s $\backslash )$ and $\backslash ($ P$-$h $\backslash )$ diagrams and indicate the state points and processes lines.

b$.$ Calculate the cooling load, in kW$.$

c$.$ Calculate the power input required for the compressor, in kW$.$

d$.$ Determine the overall COP of the refrigeration system.

e$.$ Study the effect of air volume flow rate on the cooling load, the net power input, the COP of the system.

Assumptions:

i$.$ The superheat temperature can be estimated to be $\backslash (7^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ entering the compressor.

ii$.$ The subcooling temperature can be estimated to be $\backslash (5^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ leaving the condenser.

iii. Compressor operates with isentropic efficiency of $0\mathrm{.}8.$

iv$.$ Potential and kinetic energy changes may be neglected everywhere.

v$.$ Isenthalpic expansion process in the throttling valve.

vi$.$ Neglect pressure drop in pipes, and heat losses in compressor and environment.

vii. The condensate water leaves at the average temperature of the air inlet and exit.

viii. Assume average source temperature of the air inlet and exit.