Question

Reconsider Prob. 9-60. To reduce the cost of heating the pipe, it is proposed to insulate it with enough fiberglass insulation $(k=0.035 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\( wrapped in aluminum foil \)(\varepsilon=0.1)$ to cut down the heat losses by 85 percent. Assuming the pipe temperature must remain constant at \(25^{\circ} \mathrm{C}\), determine the thickness of the insulation that needs to be used. How much money will the insulation save during this 15 -h period? Evaluate air properties at a film temperature of \(5^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) pressure. Is this a good assumption? Answers: \(1.3 \mathrm{~cm}\), \(\$ 33.40\)

Short answer

Question: Calculate the insulation thickness required to reduce heat loss by 85% and the money saved during a 15-hour period, assuming air properties at 5°C and 1 atm pressure. Discuss the assumption of the air film temperature. Answer: To find the required insulation thickness, first calculate the initial heat loss, then find the reduced heat loss, and finally, calculate the insulation thickness using the formula for heat loss through insulation. The money saved can be calculated by finding the difference in energy consumption between the initial and reduced heat losses and multiplying it by the cost per kWh. The assumption of the air film temperature at 5°C is reasonable as it allows us to calculate the properties of the surrounding air that would impact heat transfer and provides a reasonable estimate of the actual air properties in this situation.

Step-by-step solution

Calculate initial heat loss

To calculate the initial heat loss from the pipe, we will use the following formula: \(q = hA(T_s - T_\infty)\) where: - \(q\) is the heat loss (W) - \(h\) is the convection heat transfer coefficient (W/m²K) - \(A\) is the surface area of the pipe (m²) - \(T_s\) is the surface temperature of the pipe (K) - \(T_\infty\) is the temperature of the surrounding air (K) We need to find the convection heat transfer coefficient, \(h\). For the given problem we have: \(h = Nu \frac{k_f}{D}\) where: - \(Nu\) is the Nusselt number - \(k_f\) is the thermal conductivity of the fluid (W/mK) - \(D\) is the diameter of the pipe (m) The Nusselt number \(Nu\) can be calculated using the empirical correlation: \(Nu = C Re^m{Pr^n}\) where: - \(C\), \(m\), and \(n\) are constants - \(Re\) is the Reynolds number - \(Pr\) is the Prandtl number Calculate the values of \(Nu\), \(h\), and \(q\) using the provided fluid properties and constants.

Calculate reduced heat loss

We are given that the heat losses need to be reduced by 85 percent, so we need to find the reduced heat loss. We will use the following formula: \(q_{reduced} = 0.15q\) Calculate \(q_{reduced}\) using the value of \(q\) obtained in Step 1.

Calculate insulation thickness

Now, we need to find the insulation thickness needed to achieve the reduced heat loss. To do this, we will use the formula for the heat loss through the insulation: \(q_{reduced} = \frac{2 \pi k (T_s - T_\infty)}{\ln{(r_2/r_1)}}\) where: - \(k\) is the thermal conductivity of the insulation (W/mK) - \(r_1\) is the inner radius of the insulation (m) - \(r_2\) is the outer radius of the insulation (m) Rearrange the formula to solve for the insulation thickness, and then calculate the thickness: \( thickness = r_2 - r_1\)

Calculate money saved

To find the money saved during the 15-hour period, we need to calculate the difference in energy consumption between the initial and reduced heat losses. We will use the following formula: \(EnergySaved (kWh) = \frac{(q - q_{reduced}) \times Time}{1000}\) where: - \(Time\) is the duration of the 15-hour period (s) Finally, we can calculate the money saved by multiplying the energy saved by the cost per kWh: \(MoneySaved = EnergySaved \times CostPerkWh\) Calculate the money saved using the given cost per kWh.

Discuss the assumption of the air film temperature

In this problem, we are given that we must evaluate air properties at a film temperature of 5°C and 1 atm pressure. This assumption of the air film temperature is reasonable as it allows us to calculate the properties of the surrounding air that would impact heat transfer. Furthermore, as the pipe temperature is constant at 25°C and the overall temperature difference is not substantial, this assumption is likely to provide a reasonable estimate of the actual air properties in this situation.