Question

How would the answer to Prob. 11-150 be altered if we tried to approximate the heat transfer loss through the blood vessel with a fouling factor to describe physiological inhomogeneities? Assume that the fouling factor for the artery is \(0.0005\) \(\mathrm{m}^{2}\). K/W and the fouling factor for the vein is \(0.0003 \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\).

Short answer

Answer: The inclusion of fouling factors for the artery and vein increases the overall resistance to heat transfer, resulting in a lower heat transfer rate compared to the original calculation in Prob. 11-150. This demonstrates how physiological inhomogeneities can impact heat transfer loss through blood vessels.

Step-by-step solution

Understand the fouling factor

The fouling factor is a measure of the resistance to heat transfer caused by deposition or buildup of impurities or unwanted materials on heat transfer surfaces. The higher the fouling factor, the lower the overall heat transfer rate. In this exercise, we are given a fouling factor for both the artery and the vein. Including the fouling factor should thus increase the overall resistance and reduce the heat transfer rate.

Calculate the total resistance considering the fouling factors

First, the overall resistance for both the artery and vein WITHOUT fouling factors (denoted R1 and R2 respectively) should have been calculated in Prob. 11-150 using formulas from heat transfer. Now, we have to include the fouling factors in these resistances. We can do this by adding the fouling factor values directly to the calculated resistances for each blood vessel. Let \(R1^*\) and \(R2^*\) be the modified resistances for the artery and vein, including the fouling factors: $$R1^* = R1 + 0.0005 \mathrm{~m}^{-2} \cdot \mathrm{K} / \mathrm{W}$$ $$R2^* = R2 + 0.0003 \mathrm{~m}^{-2} \cdot \mathrm{K} / \mathrm{W}$$

Calculate the new overall heat transfer rate

To find the altered heat transfer rate, we need to use the formula for heat transfer considering the modified resistances \(R1^*\) and \(R2^*\), $$Q = \frac{(\Delta T)} {R_{total}}$$ where \(Q\) represents the heat transfer rate, \(\Delta T\) is the temperature difference between the artery and vein, and \(R_{total}\) is the total resistance in the system, which can be calculated as: $$R_{total} = R1^* + R2^*$$ Since we have already calculated \(R1^*\) and \(R2^*\) in Step 2, we can substitute these values in the equation above to find the new total resistance. Then, by combining this with the temperature difference between the artery and vein, which was provided in Prob. 11-150, we can determine the new heat transfer rate.

Compare the new and old heat transfer rates

Finally, we should compare the new heat transfer rate (calculated with the fouling factors) to the original heat transfer rate (calculated in Prob. 11-150). This will help us understand how the answer to Prob. 11-150 is altered when the fouling factors are considered. The difference between the two heat transfer rates will indicate the impact of physiological inhomogeneities on heat transfer loss through the blood vessel. In conclusion, by considering the fouling factors for the artery and vein, we can find the new overall resistance, and thus the new heat transfer rate, which would be lower than originally calculated in Prob. 11-150. This shows how including the fouling factors for physiological inhomogeneities would alter the answer to the original problem.