First if you are going to use 2.4 GHz i am assuming you
are going to use the ISM band. The ISM band is rectricted
in US to 30 dBm EIRP maximum. Second the link budget
is going to depend on the type of wireless link you have:
a) if the link is NOS the formula is aproximated:
Pr=Po-10nlog(d[Km]) whera A is the power at 1KM and
n is between 3.0 and 4.0
b) For LOS the formula is:
Pr=PoGtGr(lambda)^2/(4(pi)d)^2 which then became
Pr=Po+Gt+Gr-32.6-10log(d[Km))
If you need 50 microwatt= -13 dBm and asuming a G=6dB
the max distance would be (LOS)
10log(d[Km])= Po-Pr+Gt+Gr-32.6=30 dBm+6+13dBm -32.6=
16.7 dB= 43 Km
If the system is unidirectional you can use a bigger receiving
antenna to increase the distance but if the system is bidirectional
you can't put more than 30 dBm EIRP. However, you will need
some of the power for information processing. Lets say you need
about 2dBm of signal to process, lets say you need another
2 dBm for the upconverter, and the upconverter has an efficiency
of 75 %. You loss about 5 dB, then the distance will be
16.7-5= 11.7= 14 Km. The antenna is a passive device, does not
consume energy, the use of a 2 dB gain antenna does not make
sense to me.
I hope this help.
Added after 8 minutes:
I forgot. At 14 Km you will receive the average of the
signal. You will need to increase the margin in order to
have a reliable link. Assuming LOS and Rice distribution
you could expect a variability of up to 6 dB easily. If
your sensor network is not going to be miision critical
and it does not need high reliablity you can forget about it
and just tolerate retransmission. However, good engineering
justment will tell you to add at least 3 dB more for margin.
This will put the maximum distance at 8.7 dB = 7.41 Km
Added after 1 hours 31 minutes:
I made a mistake: the second formula must
include a term dependent of the frequency so
Pr=Po+Gt+Gr-32.6-10log(d[Km)) will
be Pr=Po+Gt+Gr-32.6-10log(d[Km))-10log(f(MHz))
so for 2.4 GHz 10log(2400) = 33.8 dB
If you add 33.8 dB you will have less than 1 Km
of distance. So, the project is not viable.