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Bernoulli’s Principle
- For a streamline fluid flow, the sum of the pressure (P), the kinetic energy per unit volume (ρv2/2) and the potential energy per unit volume (ρgh) remain constant.
- Mathematically:- P+ ρv2/2 + ρgh = constant
- where P= pressure ,
- E./ Volume=1/2mv2/V = 1/2v2(m/V) = 1/2ρv2
- E./Volume = mgh/V = (m/V)gh = ρgh
Derive: Bernoulli’s equation
Assumptions:
- Fluid flow through a pipe of varying width.
- Pipe is located at changing heights.
- Fluid is incompressible.
- Flow is laminar.
- No energy is lost due to friction:applicable only to non-viscous fluids.
- Mathematically: -
- Consider the fluid initially lying between B and D. In an infinitesimal timeinterval Δt, this fluid would have moved.
- Suppose v1= speed at B and v2= speedat D, initial distance moved by fluid from to C=v1Δt.
- In the same interval Δtfluid distance moved by D to E = v2Δt.
- P1= Pressureat A1, P2=Pressure at A2.
- Work done on the fluid atleft end (BC) W1 = P1A1(v1Δt).
- Work done by the fluid at the other end (DE)W2 = P2A2(v2Δt)
- Net work done on the fluid is W1 – W2 = (P1A1v1Δt− P2A2v2Δt)
- By the Equation of continuity Av=constant.
- P1A1 v1Δt - P2A2v2Δt where A1v1Δt =P1ΔV and A2v2Δt = P2ΔV.
- Therefore Work done = (P1− P2) ΔVequation (a)
- Part of this work goes in changing Kinetic energy, ΔK = (½)m (v22 – v12) and part in gravitational potential energy,ΔU =mg (h2 − h1).
- The total change in energy ΔE= ΔK +ΔU = (½) m (v22 – v12) + mg (h2 − h1). (i)
- Density of the fluid ρ =m/V or m=ρV
- Therefore in small interval of time Δt, small change in mass Δm
- Δm=ρΔV (ii)
- Putting the value from equation (ii) to (i)
- ΔE = 1/2 ρΔV (v22 – v12) + ρgΔV (h2 − h1) equation(b)
- By using work-energy theorem: W = ΔE
- From (a) and (b)
- (P1-P2) ΔV =(1/2) ρΔV (v22 – v12) + ρgΔV (h2 − h1)
- P1-P2 = 1/2ρv22 - 1/2ρv12+ρgh2 -ρgh1(By cancelling ΔV from both the sides).
- After rearranging we get,P1 + (1/2) ρ v12 + ρg h1 = (1/2) ρ v22 + ρg h2
- P+(1/2) ρv2+ρg h = constant.
- This is the Bernoulli’s equation.
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| The cells present in the Proximal Convoluted Tubule (PCT) have brush border which is capable of absorption. | Distal Convoluted Tubule (DCT) is the highly coiled part found in the medulla of the kidney. |
| URINE | |
| Proximal Convoluted Tubule (PCT) receives primary urine from glomerulus. | Distal Convoluted Tubule (DCT) receives urine from the glomerulus. |
| RESULTS IN | |
| PCT regulates excretion of urine by absorbing excess water from primary urine. | In DCT, water and salts are secreted. |
| FUNCTION | |
| A small amount of tubular secretion also takes place in the Proximal Convoluted Tubule (PCT). | The secretion of wastes of ions takes place in the Distal Convoluted Tubule (DCT). |
| OTHERS | |
| The substances which are reabsorbed into peritubular network from PCT are glucose, amino acid's, vitamin-c, potassium, Calcium, sodium chloride, 75% of water, etc. | The substances like urea, uric acid, creatinine, sodium, potassium, hydrogen ions are secreted into DCT. |
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