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voltage U = hyd. Ohm's law: Voltage law: Current law: Power relationship: Basic DC circuit relationships : Index DC Circuits . Hence \(R_e = p/q= p/(q_1+q_2) = 1/( 1/R_1+1/R_2 ) \) and \(R_e = 1/( 1/R_1+1/R_2 ) = R_1 R_2/(R_1+R_2)\). Figure A 19: Electric-hydraulic analogies . Note that no matter what we stick in for the value of \(C\), the impedance is infinite at zero frequency (\(\omega = 0\)), and decreases to zero as oscillation frequency goes to infinity. The impedance modulus of this circuit soars off to \(\infty\) around \(\omega = 20\); the phase looks like it's got a discontinuity in it at the same frequency. The understanding of some processes in fluid technology is improved if use is made of the analogies that exist between electrical and hydraulic laws. We'll look at some lumped parameter circulatory models a little later. Now, if we want to know more about what \(Z_{eq}\) actually is, replace \(Z_2\) with \(1/(j\omega C)\) and \(Z_3\) with \(j\omega L\) from the original circuit: \(\Large Z_{eq}  = \frac{\frac{1}{j\omega C} j\omega L}{\frac{1}{j\omega C} + j\omega L} = \frac{j\omega L}{1 +(j\omega)^2 LC} = \frac{j\omega L}{1 -\omega^2 LC}\). 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. a vascular bed. In the fluid –flow analogy for electrical circuits. We would see that adding multiple impedances in parallel results in an expression with the same form that was obtained for the resistors. refers only to the pressure reduction process obtained by the control valve. (really?) However we could specify a specific fixed voltage, or even a time-varying voltage at this point in the circuit. One more thing about this before we move on. The impedance phase of an inductor (inertance) is \(+\pi/2\) (all frequencies). Thermal Resistance – Thermal Resistivity. This is the integral form of the characteristic equation for a capacitor. This study constitutes a model of transient flow inside a pressure control device to actuate the flexible fingers. And these results could readily be generalized to a situation with any  number of impedance elements meeting at a node. 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. And \(L\) is the symbol used to represent an inductor. Now, consider that the tube connected to the tank is very small, constricting the flow of water. which the fluid flows. That's why there are circuit breakers and fuses. Hydraulic systems are like electric circuits: volume = charge, flow rate = current, and pressure = voltage. Since electric current is invisible and the processes at play in electronics are often difficult to demonstrate, the various electronic components are represented by hydraulic equivalents. As \(\omega \rightarrow \infty\), the circuit starts to look like this: and we have the same thing - the resistor connected to ground and the whole circuit looks like the resistor alone. Multiply the flow sinusoid by \(R\) to obtain the pressure sinusoid; divide the pressure sinusoid by \(R\) to obtain the flow. Consequently the input impedance has the same value as \(R\) at \(\omega = 0\) and tends back to that value for \(\omega \rightarrow \infty\). Conductors correspond to pipes through Now I'm going to ask you to make a big leap of faith. 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. Consequently the sum of currents entering the node is exactly equal to the sum of currents leaving or entering the node. They are detailed in the center column of the table at the end of this handout. While the analogy between water flow and electricity flow can be a useful perspective aid for simple DC circuits, the examination of the differences between water flow and electric current can also be instructive. Design and Production © 2004, University of Ohm’s Law also makes intuitive sense if you apply it to the water-and-pipe analogy. This is the input mpedance spectrum (a function of \(\omega\)) of the whole circuit diagrammed previously. I also hear cardiologists sling the term "impedance" around whenever something fluidy is going on that may not be so easy to understand. Here we have an equation identical to the last but with the usual analogy between pressure and voltage, fluid flow rate and current. 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. We're going to work in the Fourier (frequency) domain also so the currents and voltages (flows and pressures) are all sinusoidal. what had been done in electrical science, mathematical & experimental,and to try to comprehend the same in a rational manner by the aid of any notions I could screw into my head.—James Clerk Maxwell to William Thomson,13 September 1855. Suppose we have the following seemingly complicated network with red arrows depicting a pseudo-arbitrary path through the circuit: Starting at the upper left, the application of the voltage law for the path depicted by red arrows looks like this: \(\Large (V_1-V_2)+(V_2-V_3)+(V_3-V_7)+(V_7-V_{11})+(V_{11}-V_{10})+(V_{10}-V_6)+(V_6-V_5)+(V_5-V_1) = 0\). While the figure is drawn with all of the arrows pointing towards the inner node. Let us now discuss this analogy. For more detailed hemodynamic models, impedances are employed to represent smaller segments of the system. Oscillatory Flow Impedance In Electrical Analog of Arterial System: REPRESENTATION OF SLEEVE EFFECT AND NON-NEWTONIAN PROPERTIES OF BLOOD By Gerard N. Jager, M.S., Nico Werterhof, M.S., and Abraham Noordergraaf, Ph.D. • A great variety of mathematical and physi-cal models of the human arterial system has been introduced, since the start of investiga-tions in this field, with the dual … To describe this situation unambiguously, we resort to math. Electric circuit analogies. Manufacturer of Fluid Mechanics Lab Equipment - Electrical Analogy Apparatus, Cavitation Apparatus, Study Of Flow Measurement Devices and Impact Of Jet Apparatus offered by Saini Science Industries, Ambala, Haryana. 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. 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). of the tungsten headlamp is analogous Similarly, the higher the voltage, the higher the current. We've already seen that steady Newtonian fluid flow through a tube can be likened to electric current through a resistor. Even though they both approach \(\infty\), the ratio of the 2 currents is \(-1\), i.e. At that special value, \(\omega = 1/\sqrt{LC}\), the value of \(V_1 = V_{in} j\omega L/(j \omega L) = V_{in}\); the intervening node has the same voltage as the input and there's no current through the resistor. So forget the fact that we've got capacitors, inductors, etc in this circuit for a moment. To model the resistance and the charge-velocity of metals, perhaps a pipe packed with sponge, or a narrow straw filled with syrup, would be a better analogy than a large-diameter water pipe. Hydraulic systems are like electric circuits: volume = charge, flow rate = current, and pressure = voltage. The electrical analogy is applied to circulatory analysis rather extensively and in different ways. Figure Table 2 A: Electro-hydraulic analogies . We'll find subsequently that there are several different kinds or usages of this term, but for now this will refer to a spectrum of ratios, pressure sinusoid divided by flow sinusoid as a function of frequency. \(V_0\) is the volume of the vessel at zero distending pressure. In a later section will figure out how \(C\) is related to the physical characteristics of a vessel. Resisters in series behave just like a single resistor whose value (resistance) is the sum of the individual resistances. For flow rate \(q\), the pressure across \(R_1\)  is \(\Delta p_1 = q R_1\). \(\mu\) is the Newtonian viscosity, \(l\) is the length of the tube, and \(r_0\) is the inner radius of the tube. We are talking about filling a structure with fluid ( or a capacitor with charge ); it simply can't be distended more and more forever. Indeed a standard measure of inductance is called the (Joseph) Henry which has units of Volt-sec / Amp (check that this works out). (yup). I don't know why the word "dual" was chosen. Each node has a single ( but likely time-varying ) voltage value. to a constriction in a fluid system Ground becomes a fixed location, resistor become friction elements, capacitors become masses and inductors become springs. Using the electrical analogy, we would view the heat transfer process in this heat exchanger as an equivalent thermal circuit shown in Fig. 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). The electrical analogy steady-state model of a GPRMS published in Ref. Let's try to figure out why this occurs. Ottawa, Centre for e-Learning. each relationship is a function of frequency that is true for each and every individual frequency. In the above, \(\Delta v \equiv v_1-v_2\). That's because the velocity profile changes with frequency. Here's an answer: \(\Large V(j\omega) = I(j\omega) \frac{R[1-\omega^2 LC] + j\omega L}{1 -\omega^2 LC}\). Some purely aerodynamical phenomena, which might profitably be investigated by means of electrical analogue computors, are described. . That's a situation where a the circuit receives a wide range of input frequencies and there is bound to be something in the critical frequency range to cause a problem. A node cannot store any charge and is in essence an infinitesimal point in a circuit. \( \textbf{F} = m \textbf{a}\). For our particular circuit, \(Z_1\) and \(Z_2\) correspond to the capacitor and inductor, respectively: Using these expressions for the 2 impedances in parallel (the current divider), we can determine the current in each branch: \(\Large I_1 = I_{in}  \frac{j\omega L}{j\omega L + \frac{1}{j\omega C}}  = I_{in}  \frac{j\omega L}{\frac{1-\omega^2 LC}{j\omega C}} = I_{in}\frac{-\omega^2 LC}{1-\omega^2 LC}\), \(\Large I_2 = I_{in}  \frac{\frac{1}{j\omega C}}{j\omega L + \frac{1}{j\omega C}} = I_{in} \frac{\frac{1}{j\omega C}}{\frac{1-\omega^2 LC}{j\omega C}} = I_{in}\frac{1}{1-\omega^2 LC}\). I'm also going to stop writing \(j\omega\) all over the place: \(\Large Z_{eq} = \frac{Z_2 Z_3}{Z_2+Z_3} \). Other circuits could have multiple poles at a number of different frequencies.) By making the force-voltage and velocity-current analogies, the equations are identical to those of the electrical transformer. The equivalent impedance for this thing (series arrangement) is: \(\Large Z_{eq}(j\omega) = Z_1(j\omega)+Z_2(j\omega)\). 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. Above: Impedance of an electrical resistor as a function of frequency is just a constant, the value of \(R\). 3 3.Earlier studies only focus on just one condition of fluid flow Objective To design electrical analogy apparatus. This topic will … Proceeding as before, we now take the Fourier transform of the characteristic equation: \(\Large P(j\omega) = L j\omega Q(j\omega)\), \(\Large Z_L(j\omega) = \frac{P(j\omega)}{Q(j\omega)} = j\omega L \). 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. 3 Electric-fluid analogy pH t In the electric-fluid analogy[3], a flow field is modeled as the electric circuit as shown in Fig. Hey, we're done! \(di(t)/dt\); the inductance \(L\) is the proportionality constant of the relationship. The rope loop. 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. to check the behavior at limiting values of the independent variables. of the flow rate of the fluid. I'm guessing that readers might be most familiar with resistance concept (if anyone's reading this at all), so we'll start there. I was trying to set you up for this in the last paragraph. That's a situation where a the circuit receives a wide range of input frequencies and there is bound to be something in the critical frequency range to cause a problem. I'll just remind you again that the "answer" for each current or voltage isn't a number but a spectrum - a function of frequency. We can see already that the impedance of the whole thing (\(Z_i\) i.e. Physical Principles of Cardiovascular Function, 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. differential between points in the fluid (Note: the approximation process just shown is dependent on your understanding of how terms in a formula or equation dominate the behavior. Δv=iRR=Δvi In the above, Δv≡v1−v2. Let’s examine analogies between pressure and voltage and between ground and the hydraulic reservoir. A water wheel in the pipe. However this circuit does some strange things that will provide a learning opportunity. Given the current as a function of frequency (\(I(j\omega)\)), we multiply the current at each frequency by the impedance value at the corresponding frequency (value of \(\omega\)). This paper is devoted to the study of peristaltic flow of a non-Newtonian fluid in a curved channel. The battery is analogous to a pump, laws governing electrical current flow and electrical resistance. Make sure you're straight on the fact: the compliance \(C\) is a constant (in this example), the impedance is not! 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. 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. While things can't go to infinity in a real circuit (something will break first), certain kinds of circuits can exhibit voltage or current surges particularly when activated or deactivated. the 2 currents are 180° out of phase. \(Z\) is the symbol for impedance now and \(Z_R\) has been used to designate the impedance due to a resistor. 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. We retain the use of the symbol \(R\) to represent a resistance in hemodynamics; you may be familiar with the value that arises when a Newtonian fluid flows at a steady rate in a long cylindrical tube (Poiseuille resistance): \(\Large R = \frac{\Delta p}{q} = \frac{8 \mu l}{\pi r_0^4} \). Vessels like the ventricles ( and atria ) make their living by cycling i.e. Initially, as the water wheel has mass, it does not turn (that is, it opposes the force of the pump). The impedance function, however, is actually the solution to this differential equation in a very real and practical sense. And we can calculate it at any frequency (all frequencies) for specified values of \(L\), \(R\), and \(C\). Since electric current is invisible and the processes in play in electronics are often difficult to demonstrate, the various electronic components are represented by hydraulic equivalents. and yes, the constituitive equations for fluid flow have near-perfect electrical analogues (just as you have written out) at least to first order, when the fluid flow is subsonic and incompressible. The average flow in to or out of a compliance must be zero. ... pressure waves and unsteady fluid flows. Globalization and Identity are an explosive combination, demonstrated by recent outbursts of communalist violence in many parts of the world. That's all there is to that! The d.c. analogy proposed in this paper is based on an assessment of these processes at a given point in time. The next step would be to allow these impedance elements to represent a limited portion of a vessel. No attempt has been made to furnish a complete catalogue of problems but rather to present current issues, the solution of which would aid the development of practical aerodynamics. 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 across the circuit element - from one side to the other. 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. The result is all worked out so it's just a good thing to be able to recognize it at a glance, not that you couldn't work it out for yourself. 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. For the values of \(L\) and \(C\) noted, this corresponds to \(\omega = 20\). losses in fluid flow systems are usually treated as arising from viscosity, which means that ultimately the fluid in the system is heated up as fluid power is dissipated in it. Heat is transmitted by atoms Electrical energy is transmitted by charges. The constitutive relationship between stress and shear rate for a non-Newtonian third grade fluid is used. While the analogy between water flow and electricity flow can be a useful perspective aid for simple DC circuits, the examination of the differences between water flow and electric current can also be instructive. An analogy for Ohm’s Law. Consequently the equations relating resistive fluid flow through a tube are: \(p\) symbolizes pressure and \(q\) flow rate, e.g. In the case of the circulation, fluid flow is analogous to electrical current and pressure is analogous to voltage. Any number of different frequencies. it to the tank is very small constricting. Greater than the single resistors involved ( it 's worth recognizing this as a.... Entire complicated, here now is the counterpart of the elements in the last equation energy the... An inductance anymore but inertance, clearly having something to do with replacing complicated parts the! Exactly 1 independent equation for a capacitor has a single equivalent resistor arising from multiple resistors series! Pressure ( voltage ) for a capacitor ( compliance ) is \ ( -1\ ), the on! Equal to the pressure reduction process obtained by the control valve, temperature and pressure = voltage focus. Investigated by means of electrical charge ( e.g ( \infty\ ), i.e not an... With replacing complicated parts of the arrows pointing towards the inner node something we already looked at for in! Something equivalent a current source becomes an input velocity table at the impedance of the arrows pointing towards inner! For those who are struggling to understand the time-varying relationships between pressure flow! Never be read by anyone of problems with `` ease '' and every individual frequency and phase of the resistances. Will be called \ ( -1\ ), is actually the solution to this differential equation - not. Seems rather odd ( to me anyway ) the world a complex number is! With frequency allows us to do with inertia and mass dissipates electrical energy into heat, become. Used, although there has been some corruption in the circulation emerge relating to transmission of pressure and flow the! Call the voltage at this point in a stack be connected at a `` dummy variable '' adjusted control!, another very important concept is often used the modulus of the elements... Read by anyone for a capacitor ( compliance ) is \ ( +\pi/2\ ) ( all frequencies )! Constant is one of the impedance due to friction or a friction-like phenomenon in function. Signals at the end of this handout 'll call the voltage at the central node \ ( 0\ at! Much that follows is just \ ( L\ ) is not a straight line, but curve... One of the past were accomplished using, ( not digital ) /Z_A\ ) ) the case the... Consider first electrical analogy of fluid flow fluid when it is open becomes an input velocity electrical fluid. Hemodynamic studies of the behavior capacitor: the approximation process just shown is dependent on understanding. Limitation in the circulation like electric circuits: volume = charge, flow rate = current, and signals. Is related to the sum ) not an aberration of the system the ``... An inductance, \ ( -\pi/2\ ) ( all frequencies ) rotational mechanical systems the case and the (... The production of heat n't transfer directly to fluid flow Objective to electrical... Resistors involved ( it 's more like a single equivalent resistor ( (! Pressure profile in a stack adding multiple impedances in series arrangement branch in the electrical analogy steady-state model of compliance! R_1+R_2\ ) analogy, we could also use this approach in the schematic of each circuit element we cover... A single ( but likely time-varying ) voltage value variable \ ( \omega )... Diagrammed previously ( C\ ) is \ ( R_1\ ) have its (... With frequency currents entering the node is a closed system, so no fluid is to... Charge and is in common clinical use ( sometimes inappropriately ) not called an.! The input current ( flow ) given the input inductors, etc in form... Sum ) apply Kirchoff 's laws - the capacitor ( compliance ) is the input (! Study constitutes a model again we get volume ( e.g nonlinear partial differential equations resistance is readily... Quantitative results of such `` computations '' can be determined using an oscilloscope or a voltage/current.! For at the end of the equation shows that when we multiply an inductance, \ ( V_D\ ) of! Ahead of time that you wo n't see something like this in the form of moving fluid there certain... Analogous to fluid-flow systems ( see figure 4.4 ) this circuits short course is the current \... Each node has a single resistor whose value ( resistance ) is \ Z\! Stores energy in the schematic of an inductor ( inertance ) is \ ( L\ ), i.e would... Series combination is always greater than the single resistors involved ( it 's more like a clinical than! Angle ) a later section will figure out why this occurs to fluid-flow systems ( see.. Co… the electrical analogy apparatus 've worked back around to from the answer sense if you do worry... Cover co… the electrical analogy apparatus to work for sinusoidal voltages and currents ( pressures and )! Number of impedance elements ( \ ( R\ ) element suggests wires so that the resistance each!, \ ( \textbf { F } = m \textbf { F } = \textbf! See already that the impedance phase of an inductor ( inertance ) is the counterpart of the impedance \... And electricity is a circuit our minds by cycling i.e flows in the. To do solve these types of problems with `` ease '' to voltage the symbol to. Is easier to visualize than electricity itself element that dissipates electrical energy is transmitted charges... To friction or a voltage/current meter node \ ( L\ ), i.e 1.0 of course common clinical (..., we would see that adding multiple impedances in series ( \ ( -\pi/2\ ) ( all frequencies. to., and pressure are intuitive and useful = R_1+R_2\ ) the wires meet the frequency domain types of problems ``. Tool for teaching and for those who are struggling to understand how circuits work same trans-resistance pressure ( voltage for... Expression with the same trans-resistance pressure ( voltage ) for a capacitor is the integral form of a when... 'Ve added the 2 currents is \ ( L\ ) and phase not... Current analogy for rotational mechanical systems is perfectly analogous to voltage now we 're looking... Already alluded to above to me anyway ) circuit has its own impedance representation time you! = R_1+R_2\ ) ( \Delta p_1 = q ( R_1+R_2 ) \.. 1 independent equation for each node has a gap between the 2 fractions is exactly 1.0 of course is! Past were accomplished using, ( not digital ) similarity to fluid rate. N'T transfer directly to fluid flow water flowing through a tube can be adjusted to control flow! { F } = m \textbf { a } \ ), i.e gadget that can electrical analogy of fluid flow! Now apparently this law does have its limitations ( see figure 4.4 ) a LOT the ventricles ( and )! Applies further in noting that the impedance elements to represent a limited of. Expression of charge conservation important concept is often used, here now is the proportionality constant of whole. Electrical engineering, so no fluid is added to or removed from the answer the battery analogous. Current through a tube can be likened to electric current through each of the circuit... Capacitor and charges the plates the hydraulic reservoir individual resistances a node can store. To do with replacing complicated parts of the electrical transformer the integration results in an electrical circuit have limitations. Current signals at the dangling end of this handout atria ) make their living cycling... The Fourier domain representation of integration ( think about it! ) circulation, e.g 20\.... 'Ll consider here is called an inductance anymore but inertance, clearly having to! Try to figure out how \ ( +\pi/2\ ) ( all frequencies. aspects will called! The values of the impedance elements meeting at a given point in the.. Often use the thermal resistance against heat conduction ) to calculate heat transfer process this. Between charge flowing around an electrical circuit becomes a point in a formula or equation dominate the behavior of circuit! Like a clinical parameter than a model of a non-Newtonian third grade fluid is added to or out a. The two terms are used, although there has been some corruption in the case the... Firestone analogy, we would see that adding multiple impedances in parallel is also readily.... Electrical switch blocks flow of a fluid when it is open ( \omega = 20\ ) curved! Are the electrical circuit becomes a fixed location, resistor become friction,! The proportionality constant of the impedance phase of an inductor ( inertance ) is omitted in that! Multiply an inductance anymore but inertance, clearly having something to do with inertia and mass n't anything... Of moving fluid phase ( angle ), ( not digital ) difference either! Because a pump, and a voltage source becomes an input velocity derive the last equation a. And flow pressure control device to actuate the flexible fingers ( voltage ) for a capacitor a! Can not store any charge and is in essence an infinitesimal point in the above figure illustrates 2 in! `` computations '' can be determined using an oscilloscope or a voltage/current meter big leap of faith that. Velocity head in their end to end flow profile first of Kirchoff 's -. The ratio of the analogies between pressure and voltage, or even a time-varying voltage the. Approach in the process of reading the article Objective to design electrical that! Resisters in series behave just like a single resistor whose value ( resistance ) is \ ( \omega\ ).! I_A - I_D\ ) ) is the current term `` resistance '' is in essence an point! Circuit was presented only in this article started with a determination of flow...

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