Question

Glycerin is being heated by flowing between two very thin parallel 1 -m-wide and \(10-\mathrm{m}\)-long plates with \(12.5\)-mm spacing. The glycerin enters the parallel plates with a temperature \(20^{\circ} \mathrm{C}\) and a mass flow rate of \(0.7 \mathrm{~kg} / \mathrm{s}\). 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 \(3 \mathrm{~m} / \mathrm{s}\) and a temperature of \(150^{\circ} \mathrm{C}\). Determine the outlet mean temperature of the glycerin, the surface temperature of the parallel plates, and the total rate of heat transfer. Evaluate the properties for glycerin at \(30^{\circ} \mathrm{C}\) and the properties of \(\mathrm{H}_{2}\) gas at \(100^{\circ} \mathrm{C}\). Are these good assumptions?

Short answer

Answer: After following the steps provided, you should have calculated the outlet mean temperature of glycerin, the surface temperature of the parallel plates, and the total rate of heat transfer. To determine if the initial assumptions for the properties of glycerin and hydrogen gas are reasonable, compare the calculated temperatures with the assumed temperatures. If they are close, the assumptions were justified; otherwise, reiterate the calculations using revised assumptions for the fluid properties.

Step-by-step solution

Calculate the Heat Transfer Coefficient for Glycerin Flowing Between Parallel Plates

Using the Dittus-Boelter equation for a fluid flowing between parallel plates, we can find the heat transfer coefficient, \(h_g\): $$ h_g = 0.023 \times Re^{0.8} \times Pr^{0.383} $$ where \(Re\) is the Reynolds number and \(Pr\) is the Prandtl number. Note that we should evaluate the properties for glycerin at \(30^{\circ}\mathrm{C}\).

Calculate the Heat Transfer Coefficient for Hydrogen Gas Flowing Width-wise

Similar to Step 1, we can use the Dittus-Boelter equation to find the heat transfer coefficient, \(h_{H_2}\), for the hydrogen gas flowing over the plates: $$ h_{H_2} = 0.023 \times Re^{0.8} \times Pr^{0.38} $$ where \(Re\) is the Reynolds number and \(Pr\) is the Prandtl number for hydrogen gas. Note that we should evaluate the properties for hydrogen gas at \(100^{\circ}\mathrm{C}\).

Calculate the Outlet Mean Temperature of Glycerin

Using the energy balance equation and the heat transfer coefficients obtained in Steps 1 and 2, we can calculate the outlet mean temperature of glycerin (\(T_{g,o}\)): $$ \dot{m}_g c_{p,g}(T_{g,o} - T_{g,i}) = U_{overall} A (T_{H_2} - T_{g,o}) $$ where \(\dot{m}_g = 0.7\,\mathrm{kg/s}\) is the mass flow rate of glycerin, \(c_{p,g}\) is the specific heat capacity of glycerin, \(T_{g,i}= 20^{\circ}\mathrm{C}\) is the inlet temperature of glycerin, \(U_{overall}\) is the overall heat transfer coefficient, \(A\) is the heat transfer area of the plates, and \(T_{H_2} = 150^{\circ}\mathrm{C}\) is the temperature of hydrogen gas. Solve this equation for \(T_{g,o}\).

Calculate the Surface Temperature of the Parallel Plates

Using the heat transfer coefficient for glycerin calculated in Step 1, the surface temperature of the plates (\(T_s\)) can be found using the following equation: $$ h_g = \frac{\dot{m}_g c_{p,g}(T_{g,o} - T_s)}{A} $$ Solve this equation for \(T_s\).

Calculate the Total Rate of Heat Transfer

We can calculate the total rate of heat transfer, \(Q\), between the glycerin and the hydrogen gas using the overall heat transfer coefficient, the surface area, and the temperature difference between the two fluids: $$ Q = U_{overall} A (T_{H_2} - T_{g,o}) $$

Check Assumptions

Now that we have calculated the outlet mean temperature of the glycerin, the surface temperature of the parallel plates, and the total rate of heat transfer, we should check if our initial assumptions for the properties of glycerin and hydrogen gas at \(30^{\circ}\mathrm{C}\) and \(100^{\circ}\mathrm{C}\), respectively, were justified. Compare the calculated temperatures with the assumed temperatures to determine if they are reasonable. If they are not, we should iterate the calculations using revised assumptions for the fluid properties. After completing these steps, we will have the outlet mean temperature of the glycerin, the surface temperature of the parallel plates, and the total rate of heat transfer, as well as a determination of whether our assumptions were reasonable.