Large cylindrical bales of hay used to feed livestock in the

Large, cylindrical bales of hay used to feed livestock in the winter months are D = 2 m in diameter and are stored end-to-end in long rows. Microbial energy generation occurs in the hay and can be excessive if the farmer bales the hay in a too-wet condition. The thermal conductivity of baled hay is k = 0.04 W/m middot K and the ambient conditions are T_infinity = 0 degree C and h = 25 W/m^2 middot K. Determine the surface temperature and the centerline temperature for dry hay (e_gen = 1 W/m^3) and for too-wet hay (e_gen = 100 W/m^3).

Solution

Type of problem- cylinders with generation of thermal energy

From the problem-

Diameter-Do=2 m, ro=1m

K=0.004 w/m k

T ambient = 0 o C =273 K , h= 25 W/m2

Heat generation, qg =1 w/ m3     for dry hay

Heat generation, qg =100 w/ m3 for too wet hay

Solving the problem on 1m lenth basis

We know Q at surface= r2L qg

                                    =3.14 x12x1x1=3.14 W/m   for dry hay

                                    =3.14 x12x1x100=314 W/m for wet hay

We know Qat surface=h A ( Tsurface- T ambient )=25 x DL ( Tsurface- T ambient )

                        3.14=25x3.14x2x1 (Tsurface- 273)

                        Tsurface =273.02 K for dry hay

            Similarly Tsurface =279.28 K   for wet hay

Centre temperature To = Tsurface + (qg ro2 ) /(4k)

                                                =335.52 K      for dry hay

Similarly Centre temperature To    for wet hay=6529.28 K