An amount of chicken in the form of a rectangular block, 25 mm thick is roasted in a microwave heating system. The centre temperature of the chicken block is 100°C, when surrounding temperature is 30°C. The heat transfer coefficient between the chicken block and air is 15 W/m2K. The thermal conductivity of the chicken can be taken as 1 W/m.K. Calculate microwave heating capacity during steady state operation?
Hi Yogesh, this thermal problem doesn't look like a steady-state one, so thermal diffusivity value is needed.
Also, the heat transfer coefficients probably are not the same for upper and lower surfaces of the horizontally placed block of the chicken meat due to convection on its upper surface and glass or china surface contact on its lower surface.
Hi Yogesh, you can find solution of this thermal problem in Ch. III, paragraph 14.4 (for infinite time t) of the classical book Carslaw, H. S. and J. C. Jaeger. 1959. “Conduction of Heat in Solids”, 2nd edition, Oxford University Press, Oxford etc.
To calculate the microwave heating capacity during steady-state operation, we can use Fourier's law of heat conduction along with the heat transfer equation for convection. At steady state, the heat entering the chicken block via microwaves equals the heat lost from its surface to the surrounding air by convection.
For steady-state operation, the heat conduction from the center to the surface of the block balances the heat convection from the surface to the surroundings.
Step 1: Heat conduction rate:
The temperature gradient inside the block is linear under steady state. The surface temperature (Ts) of the block is needed and is given by:
q=k(Tc−Ts)/(L/2)
where L/2 is the distance from the center to the surface.
Rearranging for Ts, and knowing that q=h(Ts−T∞) from convection:
k(Tc−Ts)/(L/2)=h(Ts−T∞)
Step 2: Solve for Ts:
Combine and rearrange to express Ts:
Ts=(hT∞+(2k/L)Tc)/(h+2k/L)
Step 3: Substitute values:
Ts=(15*30+(2*1/0.025)*100)/(15+2*1/0.025)
Step 4: Heat flux q:
Once Ts is known, calculate q using either conduction or convection. Then, the microwave heating capacity (Q) is:
Q=q⋅A
where A is the total surface area of the chicken block. If needed, provide the block dimensions to calculate A.
The surface temperature of the chicken block, Ts, is approximately 88.95∘C. We can now calculate the heat flux q using either the conduction or convection equation. The heat flux, q, during steady-state operation is approximately 884.21 W/m^2.
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