Question

Saturated steam at ${100}^{\circ }\mathrm{C}$ condenses on a 2-m $\times 2-\mathrm{m}$ plate that is tilted ${40}^{\circ }$ from the vertical. The plate is maintained at ${80}^{\circ }\mathrm{C}$ by cooling it from the other side. Determine (a) the average heat transfer coefficient over the entire plate and $(b)$ the rate at which the condensate drips off the plate at the bottom. Assume wavy-laminar flow. Is this a good assumption?

Short answer

(b) What is the determined rate at which the condensate drips off the plate per unit width, and is the wavy-laminar flow assumption reasonable? (a) The calculated average heat transfer coefficient is approximately 5651 W/(m²⋅K). (b) The determined rate at which the condensate drips off the plate per unit width is approximately 0.0054 m³/s, and the wavy-laminar flow assumption is reasonable.

Step-by-step solution

Find the properties of the saturated steam

At ${100}^{\circ }\mathrm{C}$, find the properties of saturated steam using a steam table or online resources. With this information, we can obtain the density ($\rho$), dynamic viscosity ($\mu$), thermal conductivity ($k$), and latent heat of vaporization ($L$). For saturated steam at ${100}^{\circ }\mathrm{C}$, $\rho =958\mathrm{kg}/{\mathrm{m}}^3$ $\mu =2.82\times {10}^{-4}\mathrm{kg}/(\mathrm{m}\cdot \mathrm{s})$ $k=0.679\mathrm{W}/(\mathrm{m}\cdot \mathrm{K})$ $L=2.257\times {10}^6\mathrm{J}/\mathrm{kg}$

Utilize the Nusselt equation for condensation on an inclined surface

To find the average heat transfer coefficient, we need to use the Nusselt equation for condensation on an inclined surface: $Nu=\frac{hL}{k}=\left(0.943\right)\cdot {\left(\frac{g\cdot \rho (\rho -{\rho }_{\mathrm{v}})\cdot k^3\cdot L}{\mu \mathrm{\Delta }T\cdot \sin \theta }\right)}^{\frac{1}{4}}$ Where: $h$ - average heat transfer coefficient $Nu$ - Nusselt number $L$ - plate length (in this case, $2\mathrm{m}$) $g$ - acceleration due to gravity, $9.81\mathrm{m}/{\mathrm{s}}^2$ ${\rho }_{\mathrm{v}}$- vapor density (obtained from the steam table at ${100}^{\circ }\mathrm{C}$, $0.6\mathrm{kg}/{\mathrm{m}}^3$) $\mathrm{\Delta }T$ - temperature difference between the saturated steam and the plate (${100}^{\circ }\mathrm{C}-{80}^{\circ }\mathrm{C}={20}^{\circ }\mathrm{C}$) $\theta$ - angle with the vertical, ${40}^{\circ }$

Calculate the average heat transfer coefficient

By substituting our values into the Nusselt equation, we can now calculate $h$: $h=\frac{0.679\cdot \left(0.943\right)\cdot {\left(\frac{9.81\cdot 958(958-0.6)\cdot {0.679}^3\cdot 2.257\times {10}^6}{2.82\times {10}^{-4}\cdot 20\cdot \sin {40}^{\circ }}\right)}^{\frac{1}{4}}}{2}$ $h\approx 5651\mathrm{W}/({\mathrm{m}}^2\cdot \mathrm{K})$

Calculate the flow rate of the condensate

Using Reynolds analogy, we can compute the flow rate of the condensate per unit width: $R{e}_{w}L=\rho (\rho -{\rho }_{\mathrm{v}})V_w{L}^2\sin \theta /\mu$ Where: $R{e}_w$ - Reynolds number for wavy-laminar flow (from literature, $R{e}_w=1600$) $V_w$ - flow rate per unit width. Now we can solve for $V_w$: $V_w=\frac{1600\times 2.82\times {10}^{-4}}{(958)(958-0.6)(2^2)(\sin {40}^{\circ })}$ $V_w\approx 0.0054{\mathrm{m}}^3/\mathrm{s}$ The condensate drips off the plate at a rate of approximately $0.0054{\mathrm{m}}^3/\mathrm{s}$ per unit width.

Assess the wavy-laminar flow assumption

To check if the wavy-laminar flow assumption is reasonable, we need to compare the determined flow rate of the condensate ($V_w$) with the values from the literature regarding laminar and turbulent flow regimes. The given values for laminar flow are less than 30 cm/s, and turbulent flow is higher than 2 m/s. In this case, $V_w\approx 5.4\mathrm{cm}/\mathrm{s}$, which lies within the laminar flow regime, indicating that the wavy-laminar flow assumption is reasonable. In conclusion: (a) The average heat transfer coefficient over the entire plate is approximately $5651\mathrm{W}/({\mathrm{m}}^2\cdot \mathrm{K})$. (b) The rate at which the condensate drips off the plate is approximately $0.0054{\mathrm{m}}^3/\mathrm{s}$ per unit width, and the wavy-laminar flow assumption is reasonable.