 # electrical analogy of fluid flow

An inertance stores energy in the form of moving fluid. A node cannot store any charge and is in essence an infinitesimal point in a circuit. It's just a number that tells us the ratio of the voltage sinusoid to the current sinusoid (or pressure to flow) at the chosen frequency. Or, fluid flux = v . to check the behavior at limiting values of the independent variables. We've already seen that steady Newtonian fluid flow through a tube can be likened to electric current through a resistor. And the equivalent impedance of this thing? The rope loop. Electrical circuits are analogous The peaks and troughs of the voltage cycle coincide in time exactly with the peaks and troughs of the current; $$R$$ is the proportionality constant between the 2 sinusoids. Let’s examine analogies between pressure and voltage and between ground and the hydraulic reservoir. We've already seen that these correspond to $$Z_R = R$$, $$Z_L = j\omega L$$, and $$Z_C = 1/(j\omega C$$). Torque Current Analogy. each relationship is a function of frequency that is true for. 1 or any fluid flow system, it is necessary to develop the analogies between electrical quantities, and the passage of electrical current through the electrical model, and fluid flow quantities and the passage of fluid through the fluid system. though the analogy of such systems with electric systems has often been recognized and even forms a well-know,ha didactic means to explain the properties of a flow of electricity. From a mathematical standpoint, the voltage across an ideal capacitor is the integral ($$\int$$) of the current  (multiplied by a constant, $$1/C$$). A water wheel in the pipe. Now , for electric flux, think the electric field vector E in place of v. Though , electric field vector is not any type of flow, but this is a good analogy. Consequently the equations relating resistive fluid flow through a tube are: Up until now the notation has included $$\Delta p$$ (or $$\Delta v$$) to be explicit about the fact that the pressure (or voltage) is a difference, While subtle, something else has happened to this equation representing resistance; the pressure and flow got capitalized and $$j\omega$$ got stuck in all over the place. What this actually means is that a sinusoidal voltage applied across a resistor results in a sinusoidal current through the resistor that is in phase with the voltage. Amperes/sec), we'd better get a voltage. This is the input mpedance spectrum (a function of $$\omega$$) of the whole circuit diagrammed previously. A few lines up the page it was: And we can see that this thing has infinite impedance at $$\omega = 1/\sqrt{LC}$$. The equivalent resistor arising from multiple resistors in parallel is also readily determined. What does it mean? Indeed a standard measure of inductance is called the (Joseph) Henry which has units of Volt-sec / Amp (check that this works out). $$R_1/(R_1+R_2)$$. Similarly, the higher the voltage, the higher the current. Figure A 19: Electric-hydraulic analogies . First we'll cover co… Limitations of previous research 1 It lacks expansion and contraction of fluid flow apparatus 2 Flownet analysis pipes as well as spillways cannot be studied effectively. That's all there is to that! We are talking about filling a structure with fluid ( or a capacitor with charge ); it simply can't be distended more and more forever. We also see that the imaginary part is $$0$$ at $$\omega = 0$$ and tends to $$0$$ as $$\omega \rightarrow \infty$$. $$\Large I_1(j\omega) = I_{in}(j\omega) \frac{Z_2}{Z_1+Z_2}$$, $$\Large I_2(j\omega) = I_{in}(j\omega) \frac{Z_1}{Z_1+Z_2}$$. to facilitate the implementation of electrical circuits that are analogous to physical systems; In the case of the circulation, fluid flow is analogous to electrical current and pressure is analogous to voltage. Thermal-electrical analogy: thermal network 3.1 Expressions for resistances Recall from circuit theory that resistance ! Yet current flows in to the capacitor and charges the plates. each relationship is a function of frequency that is true for each and every individual frequency. Now multiplication by $$j\omega$$ in the frequency-domain is the same thing as a derivative with respect to time in the time domain: $$\Large \left[R + \frac{d}{dt} L + \frac{d^2}{dt^2} RLC\right] i(t) = \left[1 +\frac{d^2}{dt^2} LC\right] v(t)$$, $$\Large R\; i(t) + L \frac{di(t)}{dt} + RLC \frac{d^2 i(t)}{dt^2} = v(t) + LC \frac{d^2 v(t)}{dt^2}$$. In circuit-speak, the impedance is said to have a pole at $$\omega = 1/\sqrt{LC}$$ (meaning that the impedance goes to infinity. An analogy for Ohm’s Law. DERGRADUATES USING ELECTRICAL ANALOGY OF GROUNDWA-TER FLOW Murthy Kasi, North Dakota State University Murthy Kasi is currently an Environmental Engineering doctoral candidate in the Department of Civil Engineering and an Instructor in the Fluid Mechanics laboratory for undergraduates at North Dakota State University, Fargo, North Dakota, USA. So there's nothing wrong with this; or at least the math is correct and we could expect this circuit to behave something like this if we were silly enough to construct it. Now take a look at this statement and imagine how you would place the parentheses differently to show off the absurd simplicity and obvious truth of the statement. Using the voltage divider formula, the voltage $$V$$ at the intervening node is: $$\Large V = V_{in}(j\omega) \frac{Z_2}{Z_1+Z_2} = V_{in} \frac{\frac{j\omega L}{1 - \omega^2 LC}}{R +\frac{j\omega L}{1 - \omega^2 LC}} = V_{in} \frac{j\omega L}{R[1-\omega^2 LC]+j\omega L}$$. At $$\omega = 0$$ the inductor's impedance is $$j\omega L = 0$$ and the circuit reduces to this: The voltage across the capacitor is $$0$$  and the resistor is just connected to ground ($$V_1 = 0$$). That's all there is to that! To describe this situation unambiguously, we resort to math. There is a precedent for this approach in the form of a pressure profile in a stack. It's often preferable to express a complex function as a modulus (magnitude) and phase (angle). of the tungsten headlamp is analogous For flow rate $$q$$, the pressure across $$R_1$$  is $$\Delta p_1 = q R_1$$. In the schematic below, we'll call the voltage at the central node  $$V$$. So this thing: can also be represented by the following where $$Z_1$$ will correspond to the resistor, $$Z_2$$ the capacitor, and $$Z_3$$ the inductor: We just leave the type of circuit gadget out of the discussion for the time being. Their varying articulations highlight the paradox that accelerating global flows of goods, persons and images go together with determined efforts towards closure, emphasis on cultural difference and fixing of identities. The upper part of the above figure illustrates 2 resistors in series arrangement. In engineering, another very important concept is often used. How is this Used to Model the Circulation? If we were talking about resistors, equations  like $$P = Q \; R$$ and $$R = P/Q$$ wouldn't bother you. elec. Above: The impedance of a capacitor (or linear compliance) is a function of frequency even though the value of $$C$$ is a constant. However we could specify a specific fixed voltage, or even a time-varying voltage at this point in the circuit. We're just looking to separate everything the doesn't multiply $$j$$ from everything that does. Arrows depict currents flowing through each of the impedance elements ($$I_A - I_D$$). That's because the velocity profile changes with frequency. The analogy applies further in noting that the piston area ratio is perfectly analogous to the turns ratio in the transformer. So in this case the impedance spectrum of an electrical resistor is just a constant - the same value ($$R$$) at each and every frequency. The above characteristic equation for a resistor is true at all moments in time;  the voltage drop across this circuit element simply tracks the instantaneous rate of current flow with $$R$$ as the proportionality constant. View this answer. Resistance for a sinusoidal fluid flow oscillation will turn out to increase with frequency due to the fact that the velocity profile changes with frequency. I don't know why the word "dual" was chosen. Suppose that, in the fluid-flow analogy for an electrical circuit, the analog of electrical current is volumetric flow rate with units of \mathrm{cm}^{3} / \ma… Electric-hydraulic analogy. a) Frictionless pipes through which the fluid flows is analogous to conductors. $$\textbf{F} = m \textbf{a}$$. Content and Pedagogy© 2004, University of Ottawa, We've already seen that we can use it (the impedance) to calculation the voltage (pressure) given the current (flow), and vice verse. Now we're now going to replace the resistances with impedances. The impedance due to a resistance ($$Z_R$$) is ... a resistance. This is the clue that somebody has stepped in and substituted Fourier transforms in place of the pressure ($$p$$) and flow ($$q$$) from the previous equation. I'm also going to stop writing $$j\omega$$ all over the place: $$\Large Z_{eq} = \frac{Z_2 Z_3}{Z_2+Z_3}$$. All of these aspects will be addressed in due course. As a matter of fact, a significant number of physical hemodynamic studies of the past were accomplished using, (not digital). Heat is transmitted by atoms Electrical energy is transmitted by charges. The battery is analogous to a pump, Resistors are not the only kind of gadget that can appear in an electrical circuit. From a mathematical standpoint, the voltage across an ideal inductor is the derivative ($$d/dt$$) of the current  (multiplied by a constant, $$L$$). . The quantitative results of such "computations" can be determined using an oscilloscope or a voltage/current meter. Consider the pressure profile in Figure 1. (really?) $$P$$ and $$Q$$ are now pressure and flow sinusoids with an indication that they are functions of frequency ($$\omega$$) now, not time. Back to top. representing the compliance of an entire vascular bed. For any circuit, fluid or electric, which has multiple branches and parallel elements, the flowrate through any cross-section must be the same. elec. Hence the physical units work out correctly and everything on both sides of the equation is a voltage. We could also use this approach to "model" any part of the circulation, e.g. In this analogy, the pressure is the voltage, and the flow of water is the current. In terms of the voltages in the schematic, each current is equal to a voltage (difference) divided by an impedance (using the characteristic equation for each impedance element). as we study thermal and fluid systems. to fluid-flow systems (see Figure 4.4). The electronic–hydraulic analogy (derisively referred to as the drain-pipe theory by Oliver Lodge) is the most widely used analogy for "electron fluid" in a metal conductor. $$R_e$$ for a series combination is always greater than  the single resistors involved  (it's the sum). This type of circuit or physics is not going to come up in models of the circulation which is why people don't intermittantly explode. The bigger the tube, the larger the water flow at a given … The output – the result of an analog computer model is typically an electrical signal(s) – voltages and currents that vary with time. Request PDF | On Jan 1, 2019, Riccardo Sacco and others published Electric Analogy to Fluid Flow | Find, read and cite all the research you need on ResearchGate. You can't simply continue to add fluid volume to the chamber; the distending pressure would simply continue to increase until the vessel exploded. As depicted, $$V_{A-D}$$ in the above are all unknowns and we would need more information to determine the actual current through each element. Comment(0) Chapter , Problem is solved. This is going to be an important concept for understanding the behavior of a compliance in the frequency domain. The pressure-volume relationship is not a straight line, but a curve. The impedance phase of an inductor (inertance) is $$+\pi/2$$ (all frequencies). The expression for the impedance was: $$\Large \frac{V(j\omega)}{I(j\omega)} = \frac{R + j\omega L + (j\omega)^2 RLC}{1 +(j\omega)^2 LC}$$. Input impedance) is just: $$\Large Z_{i} = Z_1 + \frac{Z_2 Z_3}{Z_2+Z_3}$$. The whole thing is really just: So the first thing we'll do is replace the 2 impedances in parallel with an equivalent impedance, $$Z_{eq}$$. 3 by the analogy of basic transient equations between flow field and electric circuit The pressure P and flow rate Q correspond to 1 electric potential difference E … $$\Large q = \Sigma_{i=1}^n q_i = p \Sigma_{i=1}^n \frac{1}{R_i}$$, $$\Large \frac{p}{q} = R_e = \frac{1}{\Sigma_{i=1}^n \frac{1}{R_i}}$$. where we've added the 2 impedances in series ($$Z_1+Z_{eq}$$). We can differentiate the equation to obtain a differential form: $$\Large \frac{dv(t)}{dt} = \frac{1}{C} i(t)$$. But then we stick in limiting values, $$\tau = 0$$ and $$\tau =t$$, and end up with a function of $$t$$ (time). The final installment for this circuits short course is the current divider which was already alluded to above. The magnitude is readily determined: a complex number amounts to a right angle triangle where the 2 sides are made up of the real and imaginary parts. Yup, just like the resistors. Mobility analogies, also called the Firestone analogy, are the electrical duals of impedance analogies. The impedance is an example of a transfer function of a linear system which is the ratio of the output to the input in the frequency (Fourier) domain. In this case, the resistance is due to an entire complex network of vessels -- arteries, arterioles, microcirculation, venules, and veins. If we have a water pump that exerts pressure (voltage) to push water around a ”circuit” (current) through a restriction (), we can model how the three variables interrelate. We're going to work in the Fourier (frequency) domain also so the currents and voltages (flows and pressures) are all sinusoidal. Faculty of Engineering and Faculty of Education A capacitor has a gap between the 2 plates that's occupied by an insulator. Now we've just got 2 impedances in series, $$Z_1$$ and $$Z_{eq}$$, that can be added algebraically: The final $$Z_{eq}$$ for the whole circuit is just: $$\Large Z_{eq} = \frac{V(j\omega)}{I(j\omega)} = R + \frac{j\omega L}{1 +(j\omega)^2 LC} = \frac{R + j\omega L + (j\omega)^2 RLC}{1 +(j\omega)^2 LC} = \frac{R[1-\omega^2 LC] + j\omega L}{1 -\omega^2 LC}$$. However this circuit does some strange things that will provide a learning opportunity. Make sure you're straight on the fact: the compliance $$C$$ is a constant (in this example), the impedance is not! A current source becomes a force generator, and a voltage source becomes an input velocity. For more detailed hemodynamic models, impedances are employed to represent smaller segments of the system. Ottawa, Centre for e-Learning. Hence at least one of the currents must be negative (directed opposite the arrow) if the others are positive (in the direction of the arrow). ... pressure waves and unsteady fluid flows. A compliiance is a mechanical construct that stores energy in the form of material displacement; the term "elastic recoil" appears frequently in the medical literature but it wouldn't be a bad idea to think of a spring that can store energy in the form of tension or compression. Here's an arbitrary example problem. Generally pressure difference makes the sense. The equivalent impedance for this thing (series arrangement) is: $$\Large Z_{eq}(j\omega) = Z_1(j\omega)+Z_2(j\omega)$$. Also we are going to work for sinusoidal voltages and currents ( pressures and flows). This behavior shouldn't surprise you. The wires are assumed to have negligible resistance, inductance, or capacitance themselves, and so the value of the voltage at a node is a single value (but likely time-varying). Using the example we've started, let's see what is meant by this. Also true as $$\omega \rightarrow \infty$$ since we'll have: $$\Large V = V_{in} \frac{j\omega L}{R[1-\omega^2 LC]+j\omega L} \approx V_{in} \frac{j\omega L}{R[-\omega^2 LC]} \approx V_{in} \frac{-j}{\omega RC}$$. a line with slope $$L$$ if plotted against $$\omega$$ as shown. As the vessel portion approaches 0 length, schematic circuit elements represent a vanishingly short segment and physical units of the circuit elements change from impedance to impedance per unit of length (of vessel). I was trying to set you up for this in the last paragraph. This gives us a conceptual framework by which blood flow might be distributed and arterial pressure controlled. Electrical current is the counterpart Now I'm going to ask you to make a big leap of faith. Well, the expression can be evaluated at any (every) value of angular frequency ($$\omega$$) we choose. The modulus of the impedance is $$\omega L$$, i.e. In the study of physical hemodynamics, aspects of the circulation are often diagrammed using the very same schematic elements that are used in discussing electrical circuits. For example, we might compute the vascular resistance when trying to decide whether pulmonary hypertension is due to increased blood flow versus vascular disease (but its applicability to the pulmonary circulation is questionable -- the system is too nonlinear). While the figure is drawn with all of the arrows pointing towards the inner node, the sum of these currents must add up to ZERO. In terms of the voltages in the schematic, each current is equal to a voltage (difference) divided by an impedance (using the characteristic equation for each impedance element). If supplied as a time-domain signal, $$i(t)$$, we'd first have to determine the Fourier transform of it, $$I(j\omega)$$ (the frequency spectrum of the current signal). The equation shows that when we multiply an inductance by a current that is changing in time (e.g. And these results could readily be generalized to a situation with any  number of impedance elements meeting at a node. View a full sample. Now apparently this law does have its limitations (see the Wiki Entry for a discussion and example application) but I believe the limitations may be due to the lumped parameter schematic representation itself which does not take into account the electromagnetic fields generated by the real circuit elements. This study constitutes a model of transient flow inside a pressure control device to actuate the flexible fingers. Next we'll find the differential equation that relates the time-domain voltage and current signals at the input. Well we could have expected this by looking a little closer at the impedance of the capacitor - inductor combination before proceeding. Now this last equation is actually the question that we've worked back around to from the answer. Faculty of Engineering and Faculty of Education, Design and Production © 2004, University of Keep it in mind for what follows. Resistance is also an example of an impedance, a ratio of sinusoids (pressure over flow or  voltage over current). The equation shows that the impedance due to an inertance (or inductance) is zero at zero frequency and increases linearly with frequency. I'll warn you ahead of time that you won't see something like this in the circulation. Again this is just a commonly encountered situation, not an aberration of the rules we already know. Consider first a fluid system - this is a closed system, so no fluid is added to or removed from the system. ... source, and fluid flow. By making the force-voltage and velocity-current analogies, the equations are identical to those of the electrical transformer. Hence the total resistance can be replaced by a single resistor, $$R_e = R_1+R_2$$. a hyperbola. When we do that however, it's really meant that $$P(j\omega) = Q(j\omega)\;Z(j\omega)$$ and $$Z(j\omega)= P(j\omega)/Q(j\omega)$$, i.e. Hence $$R_e = p/q= p/(q_1+q_2) = 1/( 1/R_1+1/R_2 )$$ and $$R_e = 1/( 1/R_1+1/R_2 ) = R_1 R_2/(R_1+R_2)$$. Hence at least one of the currents must be negative (directed opposite the arrow) if the others are positive (in the direction of the arrow). Valve ) passes flow of water of impedances the heat transfer process in this heat exchanger as equivalent! An equivalent thermal circuit shown in Fig node can not store any charge and is common! Shortcomings of the impedance to the mass of the individual resistances that resistance =... This in the electrical analogy apparatus technology is improved if use is made of the were... To Power a wide range of gadgets that you use ( e.g pressure head or elevation or. 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