hw7

  • Chris Nkinthorn, 20190716

Prompt: Derive the following expressions:

  • Boundary Layer Height: $\delta(x)=\left[\frac{4 x k{l}\left(T{s a t}-T{w}\right) v{l}}{h{f g} g\left(\rho{l}-\rho_{v}\right)}\right]^{1 / 4}$

  • Heat Transfer Coefficient: $h(x)=\left[\frac{h{f g} g\left(\rho{l}-\rho{v}\right) k{l}^{3}}{4 x\left(T{s a t}-T{w}\right) v_{l}}\right]^{1 / 4}$

  • Average Heat Transfer Coefficient: $\overline{h}{L}=0.943\left[\frac{h{f \beta} g\left(\rho{l}-\rho{v}\right) k{l}^{3}}{L\left(T{s a t}-T{w}\right) v{l}}\right]^{1 / 4}$

Solution

Begin with a conservation on momentum balance, with no pressure gradient applied, to find the velocity as a function of elevation $y$.

Using this expression for the velocity, as a function of height and the pressure difference relative to vapor pressure

Take derivative to find mass change per unit length:

Need an expression for the temperature variation along the boundary layer, which we assume from the wall boundary to saturation temperature in the fluid.

The temperature distribution is important because we can relate temperature to energy. The transfer of energy in this convective mode is characterized by the heat transfer coefficient $h$. We take another average, across the temperature variation through the region of interest in $\delta$.

h_{a v}=h_{f g}+\frac{1}{\Gamma} \int_{0}^{\delta} \rho u c\left(T_{s a t}-T\right) d y\\ \begin{equation} \label{eqn:avgEnth} \begin{split} h_{a v} &=h_{f g}+\frac{\mu}{g \rho\left(\rho-\rho_{v}\right)} \frac{3}{\delta^{3}} g \rho \frac{\left(\rho-\rho_{v}\right)}{\mu} c\left(T_{s a t}-T_{w}\right) \int_{0}^{\delta}\left[\delta y-\frac{y^{2}}{2}\right]\left[1-\frac{y}{\delta}\right] d y\\ &=h_{f g}+\frac{3}{8} c\left(T_{s a t}-T_{w}\right)\\ \end{split} \end{equation}

From the temperature variation, through the heat transfer coefficient, we have an expression for energy as a function of both temperature variation and position in the boundary layer. Algebra on the differential in linear position.

Part a: Equate mass change to obtain film growth:

Solving the equation, group differentials in mass and linear position on either side, to collect mass coefficients and energy transport together.

Integrate and conduct algebra for expression, where $h{fg}’ = h{fg} + \frac 3 8 c \Delta T$ for boundary layer height:

Part c: Obtain Heat transfer coefficient (HTC), by using the two definitional expressions for $h$, Newton’s Law of Convection and equating the heat flux at the surface for conduction and convection

\label{eqn:htc} \begin{align} & h =\overbrace{\left(\frac{q}{A}\right) \frac{1}{\left(T_{s a t}-T_{w}\right)}}^\texttt{Newton's Law}=\left.\frac{k}{\delta}\right\}\texttt{Energy Balance} \rightarrow \overbrace{\frac{k}{\sqrt[4]{\frac{4\mu k\Delta T}{g\rho\left(\rho-\rho_v\right)h_{avg}} x}}}^\texttt{substitution}\\ & \boxed{h =\sqrt[4]{\frac{g \rho\left(\rho-\rho_{v}\right) k^{3} h_{f g}^{\prime}}{4 x \mu \Delta T}}} \end{align}

Part c: Average HTC comes from the average as a function of position.

Last updated