Studies on the Plant Temperature (4th Report)

The Temperature Distribution in a Fruit

Abstract

The author suggested that temperature distribution in a fruit is influenced by heat conduction from fruit surface and respiration heat originated at inner parts of a fruit.
The differential equation is
∂u/∂t=κ²∂²u/∂r²+rb² (1)
The boundary conditions are
(u)r=a=A sinωt (2)
(u)r=0=0 (3)
where u, κ², a are the temperature, the thermal diffusivity and the radius of a fruit, respectively, and rb² is the term about the resperation heat.
The required solution is thus
u(r, t)=b²(a²-r²)/6κ²+Aa√sin²hmr+sin²mr/r√sin²hma+sin²ma
×sin(ωt+tan^-1sinhmacosmacoshmrsinmr-coshmasinmasinhmrcosmr/sinhmacosmasinhmrcosmr+coshmasinmacoshmrsinmr) (4)
and the temperature at the center of a fruit is
u(o, t)=b²a²/6κ²+Aa√m/√sin²hma+sin²ma
×sin(ωt+tan^-1sinhmacosma-coshmasinma/sinhmacosma+coshmasinma) (5)
The temperature of water melon was measured and compared with calculated value from formula (4) as shown in Fig. 2.