Comet 8P/Tuttle is a Nearly Isotropic Comet (NIC), whose physical properties are poorly known and could be different from those of Ecliptic Comets (EC) owing to their different origin. Two independent observations have shown that 8P has a bilobate nucleus. Our goal is to determine the physical properties of the nucleus (size, shape, thermal inertia, albedo) and coma (water and dust) of 8P/Tuttle. We observed the inner coma of 8P with the infrared spectrograph (IRS) and the infrared camera (MIPS) of the Spitzer Space Telescope (SST). We obtained one spectrum (5-40 $mu$m) on 2 November 2007 and a set of 19 images at 24 $mu$m on 22-23 June 2008 sampling the nucleus rotational period. The data were interpreted using thermal models for the nucleus and the dust coma, and considering 2 possible shape models of the nucleus derived from respectively Hubble Space Telescope visible and Arecibo radar observations. We favor a nucleus shape model composed of 2 contact spheres with respective radii of 2.7+/-0.1 km and 1.1+/-0.1 km and a pole orientation with RA=285+/-12 deg and DEC=+20+/-5 deg. The nucleus has a thermal inertia in the range 0-100 J/K/m^2/s^0.5 and a R-band geometric albedo of 0.042+/-0.008. The water production rate amounts to 1.1+/-0.2x10^28~molecules/s at 1.6 AU from the Sun pre-perihelion, which corresponds to an active fraction of 9%. At the same distance, the $epsilon f rho$ quantity amounts to 310+/-34 cm at 1.6~AU, and reaches 325+/-36 cm at 2.2~AU post-perihelion. The dust grain temperature is estimated to 258+/-10 K, which is 37 K larger than the thermal equilibrium temperature at 1.6 AU. This indicates that the dust grains contributing to the thermal infrared flux have a typical size of 10 $mu$m. The dust spectrum exhibits broad emissions around 10 $mu$m (1.5-sigma confidence level) and 18 $mu$m (5-sigma confidence level) that we attribute to amorphous pyroxene.
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