Question

Liquid water is flowing between two very thin parallel 1-m-wide and 10 -m-long plates with a spacing of \(12.5 \mathrm{~mm}\). The water enters the parallel plates at \(20^{\circ} \mathrm{C}\) with a mass flow rate of $0.58 \mathrm{~kg} / \mathrm{s}$. The outer surface of the parallel plates is subjected to hydrogen gas (an ideal gas at \(1 \mathrm{~atm}\) ) flow width-wise in parallel over the upper and lower surfaces of the two plates. The free-stream hydrogen gas has a velocity of \(5 \mathrm{~m} / \mathrm{s}\) at a temperature of \(155^{\circ} \mathrm{C}\). Determine the outlet mean temperature of the water, the surface temperature of the parallel plates, and the total rate of heat transfer. Evaluate the properties for water at \(30^{\circ} \mathrm{C}\) and the properties of \(\mathrm{H}_{2}\) gas at \(100^{\circ} \mathrm{C}\). Is this a good assumption?

Short answer

The outlet mean temperature of the water, surface temperature of the parallel plates, and the total rate of heat transfer were determined by calculating heat transfer coefficients for water and hydrogen gas, setting up energy balance equations, and solving those equations. After calculating the outlet water temperature and surface temperature, the total rate of heat transfer was found by evaluating the energy balance equations. Finally, the assumption of properties for water at \(30^{\circ} \mathrm{C}\) and the properties of \(\mathrm{H}_{2}\) gas at \(100^{\circ} \mathrm{C}\) was evaluated and justified by comparing the results to the actual temperatures and ensuring the difference was smaller than 10%.

Step-by-step solution

From the problem statement, we can extract the following information: - width of the plates: \(w = 1 \mathrm{~m}\) - length of the plates: \(L = 10 \mathrm{~m}\) - spacing between plates: \(d = 12.5 \mathrm{~mm} = 0.0125 \mathrm{~m}\) - inlet water temperature: \(T_{\mathrm{water},\mathrm{in}} = 20^{\circ} \mathrm{C}\) - mass flow rate of water: \(\dot{m}_{\mathrm{water}} = 0.58 \mathrm{~kg/s}\) - hydrogen gas free-stream velocity: \(v_{\mathrm{gas}} = 5 \mathrm{~m/s}\) - hydrogen gas temperature: \(T_{\mathrm{gas}} = 155^{\circ} \mathrm{C}\) - pressure of hydrogen gas: \(P_{\mathrm{gas}}= 1 \mathrm{~atm}\) - properties of water: evaluate at \(30^{\circ} \mathrm{C}\) - properties of \(\mathrm{H}_{2}\) gas: evaluate at \(100^{\circ} \mathrm{C}\) #Step 2: Calculate the heat transfer coefficients for water and hydrogen gas#

First, we need to find the heat transfer coefficients for water (\(h_{\mathrm{water}}\)) and hydrogen gas (\(h_{\mathrm{gas}}\)). To do this, we will use the values of the properties of water and hydrogen at the given temperatures, such as density (\(\rho\)), specific heat (\(c_{p}\)), thermal conductivity (\(k\)), and dynamic viscosity (\(\mu\)). These properties can be found in standard reference books or online resources. #Step 3: Determine the outlet water temperature#

Now, we can set up an energy balance equation for the water flowing between the plates. The heat transfer from the water can be calculated using the heat transfer coefficient (\(h_{\mathrm{water}}\)) and the temperature difference between the water and the plates' surface (\(T_{\mathrm{surface}} - T_{\mathrm{water}}\)). The energy balance equation can be written as: $$\dot{m}_{\mathrm{water}} c_{p,\mathrm{water}} \left( T_{\mathrm{water},\mathrm{out}} - T_{\mathrm{water},\mathrm{in}} \right) = h_{\mathrm{water}} A_{\mathrm{water}} \left( T_{\mathrm{surface}} - T_{\mathrm{water},\mathrm{out}} \right)$$ where \(A_{\mathrm{water}}\) is the area of one of the plates in contact with the water, which can be calculated as \(A_{\mathrm{water}} = w L\). #Step 4: Determine the surface temperature of the parallel plates#

Similarly, the heat transfer from the hydrogen gas to the water can be calculated using the heat transfer coefficient (\(h_{\mathrm{gas}}\)) and the temperature difference between the gas and the plates' surface (\(T_{\mathrm{gas}} - T_{\mathrm{surface}}\)). The energy balance equation can be written as: $$h_{\mathrm{gas}} A_{\mathrm{gas}} \left( T_{\mathrm{gas}} - T_{\mathrm{surface}} \right) = h_{\mathrm{water}} A_{\mathrm{water}} \left( T_{\mathrm{surface}} - T_{\mathrm{water},\mathrm{out}} \right)$$ where \(A_{\mathrm{gas}}\) is the total area of the plates exposed to hydrogen gas, which is \(A_{\mathrm{gas}} = 2 w L\) (both upper and lower surfaces of the two plates). Now, we have two energy balance equations for the outlet water temperature and the surface temperature. We can solve these equations simultaneously to find the unknowns: \(T_{\mathrm{water},\mathrm{out}}\) and \(T_{\mathrm{surface}}\). #Step 5: Calculate the total rate of heat transfer#

Once we have found the outlet water temperature (\(T_{\mathrm{water},\mathrm{out}}\)) and the surface temperature (\(T_{\mathrm{surface}}\)), we can calculate the total rate of heat transfer from the hydrogen gas to the water by substituting the values we found into the energy balance equations and evaluating the terms. #Step 6: Evaluate the assumption of properties at given temperatures#

As finding values for properties of fluids such as water and H\(_2\) has limitations, some properties are listed at discrete temperature levels. The final part of this exercise is to verify if assuming the properties at given temperatures (\(30^{\circ} \mathrm{C}\) for water and \(100^{\circ} \mathrm{C}\) for H\(_2\)) is reasonable. Verify how close the assumed properties are to the real properties at the actual temperatures \(T_{\mathrm{water},\mathrm{out}}\) and \(T_{\mathrm{surface}}\). The difference should be smaller than 10% to justify the approximation used.