Question

A concentric annulus tube has inner and outer diameters of 1 in and 4 in, respectively. Liquid water flows at a mass flow rate of $396 \mathrm{lbm} / \mathrm{h}$ through the annulus with the inlet and outlet mean temperatures of \(68^{\circ} \mathrm{F}\) and \(172^{\circ} \mathrm{F}\), respectively. The inner tube wall is maintained with a constant surface temperature of $250^{\circ} \mathrm{F}$, while the outer tube surface is insulated. Determine the length of the concentric annulus tube. Assume flow is fully developed.

Short answer

Based on the provided step-by-step solution, answer the following question: Question: Calculate the length of the tube. Answer: To calculate the length of the concentric annulus tube, follow the steps outlined in the provided solution. After converting all units to SI units and performing calculations, set up the equation for annulus length (L) as: \(\frac{k}{L} = \frac{Q}{A \cdot 75}\) Solve for L to find the length of the concentric annulus tube.

Step-by-step solution

Calculate Heat Transfer through the Water

First, we need to calculate the heat transfer (\(Q\)) through the water. We can use the following equation: \(Q = m \cdot C_p \cdot \Delta T\) Where: - \(m\) is the mass flow rate (\(396 lbm / h\)) - \(C_p\) is the specific heat of water (\(1 Btu / (lbm \cdot ^{\circ}F)\)) - \(\Delta T\) is the temperature difference between inlet and outlet temperatures

Convert mass flow rate to SI units

To work with SI units, we must first convert the mass flow rate from \(lbm/h\) to \(kg/s\). Using the following conversion factors: 1 lbm = 0.453592 kg 1 h = 3600 s The mass flow rate (\(m\)) in \(kg/s\) can be calculated as: \(m = (396\frac{lbm}{h})\frac{0.453592 kg}{1 lbm}\frac{1 h}{3600 s}\)

Calculate the heat transfer

Now, we can find the heat transfer in SI units: \(Q = m \cdot C_p \cdot \Delta T\) Where: - \(C_p\) for water is \(4.186 kJ/(kg\cdot ^\circ C)\) - \(\Delta T = T_{outlet} - T_{inlet}\) Convert temperatures to Celsius: \(T_{inlet}=20^{\circ} C\) \(T_{outlet}=78^{\circ} C\) \(Q = m \cdot 4.186 \cdot (78 - 20)\)

Calculate the heat transfer coefficient and the annular area

Now we need to find the heat transfer coefficient (\(h\)) and the annular area (\(A\)). We will assume fully developed flow through the tube and constant heat transfer coefficient. From studies dealing with fully developed flow in concentric annular tubes, we know that the heat transfer coefficients tend to vary inversely with the annulus length. So we can write the heat transfer coefficient in terms of a constant (\(k\)) and the annulus length (\(L\)): \(h = \frac{k}{L}\) Next, find the annular area: \(A = \pi[(R_2^2 - R_1^2)]\) Where: - \(R_1\) is the inner radius in meters (0.5 in = 0.0127 m) - \(R_2\) is the outer radius in meters (2 in = 0.0508 m)

Use the heat transfer equation to find the tube length

Now, we can solve for the annulus length \(L\) using the heat transfer equation, relating heat transfer (\(Q\)) to the heat transfer coefficient (\(h\)), and annular area (\(A\)): \(Q = h \cdot A(L) \cdot \Delta T_{water, wall}\) Where \(\Delta T_{water, wall}\) is the temperature difference between the mean water temperature and inner tube wall temperature. Estimating \(\Delta T_{water, wall} = 128 \; ^\circ C - 53 \; ^\circ C = 75 \; ^\circ C\), we can set up the equation for annulus length (\(L\)): \(\frac{k}{L} = \frac{Q}{A \cdot 75}\) Solving for \(L\) will give us the length of the concentric annulus tube.