Fluid Mechanics for Mechanical Engineers/Energy Considerations in Internal Flows

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Losses and Energy consideration

Remember the conservation of energy equation: Template:Center top$\dot{Q}+{\dot{W}}_{shaft}+{\dot{W}}_{shear}+{\dot{W}}_{other}=\frac{\partial }{\partial t}\underset{CV}{\int }e\rho dV+\underset{CS}{\int }(e+\frac{P}{\rho })\rho U_i{n}_{i}dA$Template:Center bottom where Template:Center top$e=u+\frac{U^2}{2}+g{x}_2=u+\frac{U_i{U}_i}{2}+gz$Template:Center bottom The velocity does not have necessarily a plug profile at the inlet and outlet of a control volume and the cross sectional areas might also differ at the inlet and outlet of a CV. The variations in the cross sections of flow conduits and the velocity profiles have to be accounted to calculate the correct pressure drop due to the viscous losses. Therefore, the surface integrals has to be calculated by considering these variations: Template:Center top$\dot{Q}=\underset{CS}{\int }(e+\frac{P}{\rho })\rho U_i{n}_{i}dA=\dot{m}(u_2-u_1)+\dot{m}(\frac{P_2}{\rho }-\frac{P_1}{\rho })+\dot{m}g(z_2-z_1)+$Template:Center bottom Template:Center top$\underset{A_2}{\int }\frac{U_2^2}{2}\rho U_{2i}{n}_{2i}dA-\underset{A_2}{\int }\frac{U_1^2}{2}\rho U_{1i}{n}_{1i}dA$Template:Center bottom For the calculation of the profile effects one can use the kinetic energy coefficient $\alpha$ as follows: Template:Center top$\underset{A}{\int }\frac{U^2}{2}\rho U_i{n}_{i}dA=\alpha \dot{m}\frac{{\overline{U}}^2}{2}$Template:Center bottom For fully developed laminar velocity profile, where Template:Center top$U=2\overline{U}[1-{\frac{r}{R}}^{}2]$Template:Center bottom $\alpha =2$ For fully developed turbulent flow, due to relatively flat velocity profile, $\alpha \approx 1$ File:Losses energy.png Velocity profiles for laminar (upper) and turbulent (lower) states at the same mass flow rate Template:Center top$\dot{Q}=\dot{m}(u_2-u_1)+\dot{m}(\frac{P_2}{\rho }-\frac{P_1}{\rho })+\dot{m}({\alpha }_2\frac{{\overline{U}}_2^2}{2}-{\alpha }_1\frac{{\overline{U}}_1^2}{2})$Template:Center bottom Rearranging the energy equation: Template:Center top$\underset{A}{\underbrace{\frac{P_1}{\rho }+{\alpha }_1\frac{{\overline{U}}_1^2}{2}+g{z}_1=\frac{P_2}{\rho }+{\alpha }_2\frac{{\overline{U}}_2^2}{2}+g{z}_2}}+\underset{B}{\underbrace{(u_2-u_1)-\frac{\dot{Q}}{\dot{m}}}}$Template:Center bottom The conversion of mechanical energy (A) and the irreversible conversion and loss of mechanical energy with viscous dissipation (B) can be better distinguished. Total pressure drop Finally the total pressure drop due to viscous effects can be written as Template:Center top$\underset{MechanicalEnergy}{\underbrace{P_1+{\alpha }_1\rho \frac{{\overline{U}}_1^2}{2}+\rho g{z}_1=P_2+{\alpha }_2\rho \frac{{\overline{U}}_2^2}{2}+\rho g{z}_2}}+\underset{Pressuredropduetoviscouslosses}{\underbrace{\mathrm{\Delta }{P}_v}}$Template:Center bottom Template:Center top$\mathrm{\Delta }{P}_v=(P_1-P_2)+\rho ({\alpha }_1\frac{{\overline{U}}_1^2}{2}-{\alpha }_2\frac{{\overline{U}}_2^2}{2})+\rho g(z_1-z_2)$Template:Center bottom Major and minor pressure losses l The total pressure drop consist of major losses, which are due to frictional effects in fully developed flow in constant area tubes, and minor losses, which occur at entrances, sudden area changes, bends, elbows, fittings, valves, contraction and diffusers, etc.. Major losses in pipe flows Moody diagram The pressure drop along a pipe, which has no elevation change, can be written as: Template:Center top$\mathrm{\Delta }{P}_v=(P_1-P_2)$Template:Center bottom Nondimensionalizing with the kinetic energy per unit mass of the flow: Template:Center top$\zeta =\frac{\mathrm{\Delta }{P}_v}{0.5\rho {\overline{U}}^2}=\frac{L}{D}f(Re,\frac{e}{D})$Template:Center bottom where $\zeta$ is the loss coefficient and f is the friction factor. Non-circular ducts For non circular ducts, which appears in air conditioning, heating and ventilating applications, an equivalent diameter is calculated so that correlations for circular pipes can be utilized. This equivalent diameter is called as hydraulic diameter: Template:Center top$D_h=\frac{4\times Area}{Perimeter}=\frac{4A}{P}$Template:Center bottom Thus for a rectangular duct with a width of b and height of h: Template:Center top$D_h=\frac{4bh}{2(b+h)}$Template:Center bottom And for a square duct Template:Center top$D_h=h$Template:Center bottom Minor losses For the other components which might cause pressure drop, the loss coefficient (resistance coefficient) can similarly be defined: Template:Center top$\zeta =\frac{\mathrm{\Delta }{P}_v}{0.5\rho {\overline{U}}^2}$Template:Center bottom However, for the vast variety of geometries, it is not possible to obtain relations as for the pipe flows. Thus, $\zeta$ for each component is measured and documented. Fittings Valve Loss coefficient of standard components File:Elbow 1.png Elbow geometry File:Loss energy in valves.svg Range of loss coefficients for different valves and flaps File:Resistance Coefficient 2.png Range of loss coefficients for circular bellmouth inlet(collector) File:Losses energie 8.png The loss coeffecicent data for various elbow dimensions File:Resistance Coefficient 1.png Range of loss coefficients for the entrance into a straight tube of constant cross section File:Resistance Coefficient 3.png Range of loss coefficients for sudden expansion of a flow having a uniform velocity distribution Calculation of total loss in a single pipe system A single pipe system may have many minor losses. Since all are correlated with $0.5\rho {\overline{U}}^2$, they can be summed into a single total system loss if the pipe has constant diameter: Template:Center top$\mathrm{\Delta }{P}_{total}=\mathrm{\Delta }{P}_{pipe}+\sum \mathrm{\Delta }{P}_{components}=$Template:Center bottom Template:Center top$0.5\rho {\overline{U}}^2[\frac{L}{D}f+\sum _i{\zeta }_i]$Template:Center bottom However, we must sum the losses separately, if the pipe size changes, i.e. mean velocity changes. Multiple path system File:Losses energie11 2.png The solution for pipe network problems is often carried out by use of node and loop equations similar in many ways to that done in electrical circuits. The net flow into any junction must be zero. The net head loss around any closed loop must be zero. At each junction there is one single pressure. All head losses must satisfy the major and minor-loss friction correlations. In such problems the pipes and components might have different areas, thus pressure loss at each pipe and component should be calculated separately. Moreover, there can be components with unknown properties. For such cases it is found useful to formulate the pressure loss as a function of the flow rate: Template:Center top$\mathrm{\Delta }P=\zeta 0.5\rho {\overline{U}}^2=\underset{K}{\underbrace{\zeta \frac{0.5\rho }{A^2}}}{Q}^2=K{Q}^2$Template:Center bottom Resultant loss coefficients, Kr, for serial and parallel connections Serial connection: Flow rate is the same for all components File:Losses energie12.png Template:Center top$K_r=K_1+K_2$Template:Center bottom Template:Center top$\mathrm{\Delta }{P}_r=K_r{Q}^2=\mathrm{\Delta }{P}_1+\mathrm{\Delta }{P}_2=(K_1+K_2)Q^2$Template:Center bottom Parallel connection: Pressure drop is the same inlet and outlet junctions File:Losses energie13.png Template:Center top$K_r=\frac{K_1{K}_2}{K_1+K_2+2\sqrt{K_1{K}_2}}$Template:Center bottom Template:Center top$\mathrm{\Delta }{P}_r=K_r{Q}_{total}^2=K_1{Q}_1^2=K_2{Q}_2^2$Template:Center bottom Template:Center top$Q_{total}=Q_1+Q_2$Template:Center bottom Design of a dry powder inhaler The utilization of the explained concepts will be demonstrated in the class by explaining the design method of a dry powder inhaler. The basic steps of the design are as follows: File:Designpablo.png Schematics for Inhaler flow Make a conceptual design Decide pressure drop-flow rate characteristic of the inhaler (Inhaler patient matching) Dimension each component of the inhaler to ensure Pressure drop-flow rate characteristic Full emptying of the blister Sufficient dispersion of the powder composition An overview of dry powder inhaler design could be found in in the following publication Reference