8:2 Conservation of Energy - Examples, Differential Representation

Overview

Derek Elsworth connects conservation of energy and mass to practical fluid-flow models, from pump-and-reservoir calculations to groundwater equations. He reviews Bernoulli head accounting, solves setup steps for two pump problems, and derives how differential mass conservation combined with Darcy's law produces pressure- and head-based flow equations used in porous-media engineering.

Key takeaways

For a dam with negligible surface velocity, atmospheric pressure at both reservoirs, and no head loss, power output is the water unit weight multiplied by flow rate and elevation drop.
A pump operating between large reservoirs must supply the elevation rise plus friction losses; expressing velocity as Q/A makes the flow-rate calculation an algebraic equation.
Pump head is positive when the pump adds energy and negative when the device acts as a turbine; frictional head loss remains positive in the upstream-to-downstream balance.
Darcy's law combined with differential mass conservation yields the pressure-diffusion form used for flow through porous media.
Groundwater models often express storage and flow in terms of specific storage, hydraulic head, and hydraulic conductivity, while petroleum and geothermal models commonly use pressure, compressibility, permeability, and viscosity.
Micro-CT pore reconstruction can turn otherwise tiny well cuttings into computational permeability estimates by mapping pore geometry and simulating flow through it.

Chapters

0:00 Fluid Simulations: From Ingrain Pore Imaging to CGI Water

Ingrain's proposed workflow used micro-CT scans of small well cuttings to reconstruct pore geometry and numerically estimate permeability.
Pore-scale models impose zero-velocity conditions at solid boundaries and calculate flow under an applied pressure gradient, including simultaneous gas and water flow.
CGI water simulations represent fluids as particles or cells and apply motion equations to reproduce jets, waves, and water striking a lighthouse.
Grid-based groundwater models use the same balance logic: inflow, outflow, and accumulation, with Darcy's law describing flow through rock.

8:30 Exam Review: Practice the Covered Momentum Problems

Elsworth directs students to the online test review and says it identifies the examinable material.
The review covers free jets and conservation of linear and rotational momentum; he notes that a parabolic-jet problem was not worked in class.
Past questions are presented as useful practice because they repeat prior problem formats with changed numbers.

10:09 Energy Equation Recap: Enthalpy, Heat, and Mechanical Power

The energy balance applies the Reynolds transport theorem to mass times total energy, including elevation, velocity, pressure, and internal energy.
Pressure and internal energy are grouped as enthalpy, expressed as energy per unit mass.
The resulting balance resembles Bernoulli's equation but includes net heat transfer and mechanical power in or out of the system.
Power is a rate of energy transfer, measured in watts, equivalent to joules per second or newton-metres per second.

13:15 Bernoulli Head Accounting: Pumps, Turbines, and Losses

The upstream-to-downstream head balance includes elevation, pressure, velocity, pump head, and head loss.
Pump head is positive when a pump adds energy; a negative value indicates a turbine extracting energy.
Head losses from friction are positive and oppose fluid motion.
Pump head relates power to volumetric flow rate through power divided by flow rate times fluid unit weight.

15:13 Dam Power: Converting Reservoir Head Drop into Output

For large reservoirs, surface velocities are negligible and both free surfaces are at atmospheric pressure.
With negligible losses, the available dam power is the elevation drop multiplied by volumetric flow rate and water unit weight.
Annual runoff reaching a reservoir after evaporation and transpiration helps estimate the flow available for generation and the dam's power rating.

20:43 Pump Between Reservoirs: Solving for Flow from a Pump Curve

The example transfers water between large reservoirs with a 100-foot elevation rise and a 4-inch-diameter pipe.
The specified friction loss is a coefficient of 20 multiplied by average velocity squared over 2g.
A linear pump curve falls from 200 feet of head at zero flow to zero head at 2 cubic feet per second.
Using the large-reservoir assumptions, Elsworth sets pump head equal to the 100-foot elevation rise plus friction loss, then substitutes velocity as Q/A to obtain a quadratic in Q.

31:08 Gasoline in Two Pipes: Pressure Difference from Energy Balance

The example uses gasoline of density 680 kilograms per cubic metre, a flow rate of 0.12 cubic metres per second, and two pipes with different diameters.
Continuity gives equal volumetric flow through both pipes, so each average velocity is calculated as Q/A.
The head loss is specified as 20 times the square of the velocity in pipe 1 divided by 2g; the elevation change is 3 metres.
A 20-kilowatt pump contributes head through power divided by Q times density and gravity, allowing the Bernoulli balance to determine the pressure difference.

41:33 Differential Mass Conservation for a Small Control Volume

Elsworth derives a local mass balance by tracking flow into and out of a differential cube with dimensions dx, dy, and dz.
For the x-direction faces, the net flux is represented by the velocity difference multiplied by face area dy times dz.
Dividing the balance by the control-volume size converts the finite flux difference into a spatial velocity derivative.
Extending the same balance across all directions gives the differential continuity framework used in fluid-flow models.

47:52 Darcy's Law Turns Continuity into a Pressure-Flow Equation

For water moving through a sand-filled cylinder, Darcy's law relates velocity to permeability, pressure gradient, and fluid viscosity.
The negative sign in Darcy's law reflects flow from high to low pressure.
Substituting Darcy velocity into the differential mass balance produces the basis of groundwater-flow equations.
At steady state, the transient storage contribution vanishes, leaving a spatial pressure equation.

51:45 Storage and Compressibility in Petroleum and Groundwater Models

Fluid compressibility links changes in density to changes in pressure; Elsworth relates the density response to a compressibility coefficient.
In petroleum and geothermal settings, the pressure-based flow equation combines pressure change over time with permeability divided by viscosity times the second spatial pressure derivative.
Groundwater models commonly express the analogous equation using specific storage, hydraulic head, and hydraulic conductivity.
Hydraulic head includes pressure head and elevation; the velocity-head term is often neglected in slow groundwater flow.

55:00 Continuity, Darcy Head Loss, and Flow Through Porous Media

For steady flow with constant density, the three-dimensional mass balance reduces to the sum of velocity derivatives in the x, y, and z directions equalling zero.
The equation follows from balancing mass flux across the faces of a differential volume, rather than from a separate empirical groundwater rule.
In a sand-filled pipe of constant area, Bernoulli's elevation and velocity terms cancel between endpoints, leaving pressure loss to drive flow.
That pressure loss is represented as head loss and, through Darcy's law, extends the pipe-flow balance to porous-media flow in three dimensions.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Derek Elsworth.