In cryogenic surgery, a small spherical probe is brought into contact with the diseased tissue which is frozen and thereby destroyed. One such probe can be modeled as a 3-mm diameter sphere whose surface is maintained at 240 K when the surrounding tissue is at 314 K. During the surgical procedure, a thin layer of tissue freezes around the probe at a temperature of 273 K. Assuming that the thermal conductivity of a frozen tissue is 1.5 W/(m K) and the heat transfer mechanism at the surface is described by the effective convective coefficient of 50 W/(m2 K), estimate the thickness of frozen tissue formed during a 30 min long operation.

Question

GIVEN

Diameter of sphere(D1)= 3 mm, r1=0.0015 m

Probe surface temperature(Tp)= 240 K

Surrounding tissue temperature(Tt)= 314 K

Tissue freezing temperature(Tf)= 273 K

Thermal conductivity of frozen tissue(kt)= 1.5 W/(m K)

Effective convective coefficient (h?)=50 W/(m2 K)

Operation time (t)= 30 min

FIND

Thickness of a frozen tissue formed

ASSUMPTIONS

1 dimensional Steady state conditions persist

Negligible contact resistance between probe and frozen tissue.

Constant properties.

SKETCH

Answer 1

The thermal circuit diagram for the given as follows



The thickness of tissue in 30 mins is obtained by energy balance that is when rate of heat loss through conduction is equal to rate of heat loss through convection, which means



If r2 is the radius of frozen tissue at 30 mins

The convection heat loss



Conduction heat loss is given by



Equating above two equations we get



Solving the above equation, we get r2=0.007 m or 7 mm.

Thus the thickness of frozen tissue=r2-r1= 5.5 mm