Answer:
0.0371 kg/s.m
Explanation:
From the given information, let's have an imaginative view of the semi-cylinder; (The image is shown below)
Assuming the base surface of both ends of the cylinder is denoted by:
![A_1 \ and \ A_2](https://tex.z-dn.net/?f=A_1%20%20%5C%20and%20%20%20%5C%20A_2)
Thus, using the summation rule, the view factor
and
is as follows:
![F_{11}+F_{12}=1](https://tex.z-dn.net/?f=F_%7B11%7D%2BF_%7B12%7D%3D1)
Let assume the surface (1) is flat, the ![F_{11} = 0](https://tex.z-dn.net/?f=F_%7B11%7D%20%3D%200)
Now:
![0+F_{12}=1](https://tex.z-dn.net/?f=0%2BF_%7B12%7D%3D1)
![F_{12}=1](https://tex.z-dn.net/?f=F_%7B12%7D%3D1)
However, using the reciprocity rule to determine the view factor from the dome-shaped cylinder
to the flat base surface
; we have:
![A_2F_{21} = A_{1}F_{12} \\ \\ F_{21} = \dfrac{A_1}{A_2}F_{12}](https://tex.z-dn.net/?f=A_2F_%7B21%7D%20%3D%20A_%7B1%7DF_%7B12%7D%20%5C%5C%20%5C%5C%20F_%7B21%7D%20%3D%20%5Cdfrac%7BA_1%7D%7BA_2%7DF_%7B12%7D)
Suppose, we replace DL for
and
=
Then:
![F_{21} = \dfrac{DL}{(\dfrac{\pi D}{2}) L} \times 1 \\ \\ =\dfrac{2}{\pi} \\ \\ =0.64](https://tex.z-dn.net/?f=F_%7B21%7D%20%3D%20%5Cdfrac%7BDL%7D%7B%28%5Cdfrac%7B%5Cpi%20D%7D%7B2%7D%29%20L%7D%20%5Ctimes%201%20%5C%5C%20%5C%5C%20%20%3D%5Cdfrac%7B2%7D%7B%5Cpi%7D%20%5C%5C%20%5C%5C%20%20%3D0.64)
Now, we need to employ the use of energy balance formula to the dryer.
i.e.
![Q_{21} = Q_{evaporation}](https://tex.z-dn.net/?f=Q_%7B21%7D%20%3D%20Q_%7Bevaporation%7D)
But, before that; let's find the radian heat exchange occurring among the dome and the flat base surface:
![Q_{21}= F_{21} A_2 \sigma (T_2^4-T_1^4) \\ \\ Q_{21} = F_{21} \times \dfrac{\pi D}{2} \sigma (T_2^4 -T_1^4)](https://tex.z-dn.net/?f=Q_%7B21%7D%3D%20F_%7B21%7D%20A_2%20%5Csigma%20%28T_2%5E4-T_1%5E4%29%20%5C%5C%20%5C%5C%20Q_%7B21%7D%20%3D%20F_%7B21%7D%20%5Ctimes%20%5Cdfrac%7B%5Cpi%20D%7D%7B2%7D%20%5Csigma%20%28T_2%5E4%20-T_1%5E4%29)
where;
![\sigma = Stefan \ Boltzmann's \ constant](https://tex.z-dn.net/?f=%5Csigma%20%3D%20Stefan%20%5C%20Boltzmann%27s%20%5C%20constant)
![T_1 = base \ temperature](https://tex.z-dn.net/?f=T_1%20%3D%20base%20%5C%20temperature)
![T_2 = temperature \ of \ the \ dome](https://tex.z-dn.net/?f=T_2%20%3D%20temperature%20%20%5C%20of%20%20%5C%20the%20%20%5C%20dome)
∴
![Q_{21} = 0.64 \times (\dfrac{\pi}{2}\times 1.5) \times 5.67 \times 10^4 \times (1000^4 -370^4)\\ \\ Q_{21} = 83899.15 \ W/m](https://tex.z-dn.net/?f=Q_%7B21%7D%20%3D%200.64%20%5Ctimes%20%28%5Cdfrac%7B%5Cpi%7D%7B2%7D%5Ctimes%201.5%29%20%5Ctimes%205.67%20%5Ctimes%2010%5E4%20%5Ctimes%20%281000%5E4%20-370%5E4%29%5C%5C%20%5C%5C%20Q_%7B21%7D%20%3D%2083899.15%20%5C%20W%2Fm)
Recall the energy balance formula;
![Q_{21} = Q_{evaporation}](https://tex.z-dn.net/?f=Q_%7B21%7D%20%3D%20Q_%7Bevaporation%7D)
where;
![Q_{evaporation} = mh_{fg}](https://tex.z-dn.net/?f=Q_%7Bevaporation%7D%20%3D%20mh_%7Bfg%7D)
here;
= enthalpy of vaporization
m = the water mass flow rate
∴
![83899.15 = m \times 2257 \times 10^3 \\ \\ m = \dfrac{83899.15}{ 2257 \times 10^3 }\\ \\ \mathbf{m = 0.0371 \ kg/s.m}](https://tex.z-dn.net/?f=83899.15%20%3D%20m%20%5Ctimes%202257%20%5Ctimes%2010%5E3%20%20%5C%5C%20%5C%5C%20%20m%20%3D%20%5Cdfrac%7B83899.15%7D%7B%202257%20%5Ctimes%2010%5E3%20%7D%5C%5C%20%5C%5C%20%5Cmathbf%7Bm%20%3D%200.0371%20%5C%20kg%2Fs.m%7D)