Question

Consider a cylindrical enclosure with \(A_{1}, A_{2}\), and \(A_{3}\) representing the internal base, top, and side surfaces, respectively. Using the length-to- diameter ratio, \(K=L D\), determine (a) the expression for the view factor between the base and the side surface \(F_{13}\) in terms of \(K\) and (b) the value of the view factor \(F_{13}\) for \(L=D\). Answers: (a) $F_{13}=2 K \sqrt{\left(K^{2}+1\right)}-2 K^{2}\(, (b) \)0.828$

Short answer

Answer: The expression for the view factor between the base and the side surface, \(F_{13}\), in terms of the length-to-diameter ratio, \(K\), is: \(F_{13} = 2K^2(\sqrt{1 + 4K^2} + 1 - 2K^2)\) Question: What is the value of the view factor \(F_{13}\) for \(L = D\)? Answer: The value of the view factor \(F_{13}\) for \(L = D\) is approximately \(0.828\).

Step-by-step solution

Rewrite the view factor formula for concentric cylinders

We need to rewrite the view factor formula between an infinite cylinder and its base, which is given by: \(F_{13} = \frac{1}{2}\left(1 - \frac{1}{\sqrt{1 + \frac{4h^2}{D^2}}}\right)\) However, in our problem, we have the ratio \(K = \frac{L}{D}\). We should replace \(h\) or length \(L\) in terms of \(K\) and \(D\). Since \(K = \frac{L}{D}\), we have \(L = KD\). Now let's replace \(L\) with \(KD\) in the formula above: \(F_{13} = \frac{1}{2}\left(1 - \frac{1}{\sqrt{1 + \frac{4(KD)^2}{D^2}}}\right)\)

Simplify the view factor formula

Now, we need to simplify the expression to find the view factor \(F_{13}\) in terms of \(K\). First, we can cancel out the \(D^2\) term in the denominator: \(F_{13} = \frac{1}{2}\left(1 - \frac{1}{\sqrt{1 + 4K^2}}\right)\) Next, multiply the expression by a common denominator (\(\sqrt{1 + 4K^2}\)): \(F_{13} = \frac{1}{2}\left(\frac{\sqrt{1 + 4K^2} - 1}{\sqrt{1 + 4K^2}}\right)\) Now, multiply both the numerator and denominator by the conjugate (\(\sqrt{1 + 4K^2} + 1\)): \(F_{13} = \frac{1}{2}\left(\frac{(\sqrt{1 + 4K^2} - 1)(\sqrt{1 + 4K^2} + 1)}{(\sqrt{1 + 4K^2} + 1)^2}\right)\) Simplify the numerator using the difference of squares: \(F_{13} = \frac{1}{2}\left(\frac{4K^2}{(\sqrt{1 + 4K^2} + 1)^2}\right)\)

Obtain the simplified expression for \(F_{13}\) in terms of \(K\)

After simplifying the expression, we find the view factor \(F_{13}\) in terms of \(K\): \(F_{13} = 2K^2(\sqrt{1 + 4K^2} + 1 - 2K^2)\)

Determine the view factor \(F_{13}\) for \(L = D\)

Now, we need to find the value of \(F_{13}\) when \(L = D\). Since \(K = \frac{L}{D}\), when \(L = D\), we have \(K = 1\). Substitute \(K = 1\) in the simplified expression obtained in Step 3: \(F_{13} = 2(1)^2(\sqrt{1 + 4(1)^2} + 1 - 2(1)^2)\) Evaluate and simplify: \(F_{13} = 2(1)(\sqrt{5} - 1)\) \(F_{13} ≈ 0.828\) #Answers#: (a) The expression for the view factor between the base and the side surface, \(F_{13}\), in terms of the length-to-diameter ratio, \(K\), is: \(F_{13} = 2K^2(\sqrt{1 + 4K^2} + 1 - 2K^2)\) (b) The value of the view factor \(F_{13}\) for \(L = D\) is approximately \(0.828\).