Consider a design for a nuclear reactor using natural-convection heating of liquid bismuth as shown. The reactor is to be constructed of parallel vertical plates 1.8 m tall and 1.2 m wide, in which heat is generated uniformly. Estimate the maximum possible heat dissipation rate from each plate if the average surface temperature of the plate is not to exceed 870°C and the lowest allowable bismuth temperature is 315°C.

Question

GIVEN

• Vertical plates with uniform heat generation in bismuth

• Plate height (L) = 1.8 m

• Plate width (w) = 1.2 m

• Maximum average surface temperature (Ts) = 870°C

• Minimum bismuth temperature (T?) = 315°C

FIND

• Maximum possible heat dissipation rate (q) from each plate

ASSUMPTIONS

• Steady state

• Free convection only

• Edge effects are negligible

SKETCH



PROPERTIES AND CONSTANTS

for bismuth at the mean temperature of (592.5°C)

Thermal conductivity (k) = 15.58 W/(m K)

Kinematic viscosity (?) = 0.991 × 10–7 m2/s

Prandtl number (Pr) = 0.010 Also Density at 538°C (?538) = 9739 kg/m3 Density at 649°C (?649) = 9611 kg/m3

To find the thermal expansion coefficient (?)

Answer 1

The Grashof number based on the vertical length of the plate is



The average Nusselt number for a vertical plate in liquid metal for Gr > 109



Solving for the heat transfer coefficient



The maximum rate of heat transfer from both sides of the plate is given by