Question

The National Sanitation Foundation (NSF) standard for commercial warewashing equipment (ANSL/NSF 3) requires that the final rinse water temperature be between 82 and \(90^{\circ} \mathrm{C}\). A shell-and-tube heat exchanger is to heat \(0.5 \mathrm{~kg} / \mathrm{s}\) of water $\left(c_{p}=4200 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\( from 48 to \)86^{\circ} \mathrm{C}$ by geothermal brine flowing through a single shell pass. The heated water is then fed into commercial warewashing equipment. The geothermal brine enters and exits the heat exchanger at 98 and \(90^{\circ} \mathrm{C}\), respectively. The water flows through four thin-walled tubes, each with a diameter of $25 \mathrm{~mm}$, with all four tubes making the same number of passes through the shell. The tube length per pass for each tube is \(5 \mathrm{~m}\). The corresponding convection heat transfer coefficients on the outer and inner tube surfaces are 1050 and $2700 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, respectively. The estimated fouling factor caused by the accumulation of deposit from the geothermal brine is $0.0002 \mathrm{~m}^{2} . \mathrm{K} / \mathrm{W}$. Determine the number of passes required for the tubes inside the shell to heat the water to \(86^{\circ} \mathrm{C}\), within the temperature range required by the ANIS/NSF 3 standard.

Short answer

The required number of passes is 3.

Step-by-step solution

Calculate the heat transfer rate required

To calculate the heat transfer rate required, we can use the following equation: \(q = m_w c_{p_w} \Delta T_w\), where \(m_w\) is the mass flow rate of water, \(c_{p_w}\) is the specific heat capacity of water, and \(\Delta T_w\) is the temperature difference for the water. Given that \(m_w = 0.5 \mathrm{~kg} / \mathrm{s}\), \(c_{p_w} = 4200 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), and the water temperature needs to increase from \(48^{\circ} \mathrm{C}\) to \(86^{\circ} \mathrm{C}\), we can calculate \(\Delta T_w\) as: \(\Delta T_w = 86 - 48 = 38^{\circ} \mathrm{C}\) Now, we can calculate the heat transfer rate \(q\) as: \(q = 0.5 \cdot 4200 \cdot 38 = 79800 \mathrm{~W}\)

Calculate the overall heat transfer coefficient

Next, we need to calculate the overall heat transfer coefficient (U). We can use the following equation for this: \(1/U = 1/h_{i} + \frac{R_{f}}{h_{o}}\) Given that \(h_i = 2700 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), \(h_o = 1050 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and the fouling factor \(R_f= 0.0002 \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\). Now, we can calculate the overall heat transfer coefficient U as: \(1/U = 1/2700 + 0.0002/1050\) \(U = \frac{1}{\frac{1}{2700} + \frac{0.0002}{1050}} = 694.1 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\)

Calculate the logarithmic mean temperature difference (LMTD)

We can calculate the LMTD using the following equation: \(LMTD=\frac{(T_{h,i}-T_{c,o})-(T_{h,o}-T_{c,i})}{\ln \left[\frac{(T_{h,i}-T_{c,o})}{(T_{h,o}-T_{c,i})}\right]}\) Given that \(T_{h,i}=98^{\circ} \mathrm{C}\), \(T_{h,o}=90^{\circ}\mathrm{C}\), \(T_{c,i}=48^{\circ}\mathrm{C}\) and \(T_{c,o}=86^{\circ}\mathrm{C}\), we can calculate LMTD as: \(LMTD=\frac{(98-86)-(90-48)}{\ln \left[\frac{(98-86)}{(90-48)}\right]}= 24.68^{\circ} \mathrm{C}\)

Calculate the total heat transfer area required

The total heat transfer area required can be calculated using the following equation: \(A = \frac{q}{U \times LMTD}\) By plugging in the values we obtained earlier, the total heat transfer area is calculated as: \(A = \frac{79800}{694.1 \times 24.68} = 4.42 \mathrm{m}^2\)

Calculate the heat transfer area per pass

The heat transfer area per pass is given by: \(A_{pass} = \pi D L\) Given that the tube diameter \(D=0.025 \mathrm{m}\) and the tube length per pass \(L=5 \mathrm{m}\), we can calculate the heat transfer area per pass as: \(A_{pass} = \pi (0.025)(5) = 0.3927 \mathrm{m}^2\)

Calculate the required number of passes

Since there are four tubes in the heat exchanger, the total heat transfer area per pass for all four tubes is: \(A_{4 tubes} = 4 \times A_{pass} = 4 \times 0.3927 = 1.5708 \mathrm{m}^2\) To calculate the required number of passes, divide the total heat transfer area required by the heat transfer area per pass for all four tubes: \(N = \frac{A}{A_{4 tubes}} = \frac{4.42}{1.5708} = 2.81\) Since the number of passes must be an integer, round up to the nearest whole number to obtain the minimum number of passes required: \(N = 3\) passes Hence, the tubes inside the shell need to make 3 passes to heat the water to \(86^{\circ} \mathrm{C}\), within the temperature range required by the ANIS/NSF 3 standard.