A patient in the hospital is receiving a saline solution through a needle in the

Question

A patient in the hospital is receiving a saline solution through a needle in their arm. The needle is 2.60 cm long, horizontal, with one end in a vein in the arm, and the other end attached to a wide tube that extends down from the bag of solution, which hangs from a pole so that the fluid level is 85.0 cm above the needle. The inner radius of the needle is 0.200 mm. The top of the fluid is exposed to the atmosphere, and the flow rate of the fluid (which has a density of 1025 kg/m3 and a viscosity of 0.0010 Pa·s) through the needle is 0.200 L/h. What is the average gauge pressure inside the vein where the needle is? Use g = 9.8 m/s2.

___ Pa

Explanation / Answer

to find the pressure at the top of needle

Patm + rho g h + 0 = P + 0 + rho v^2 /2

Q = A v

0.2 x 10^-3 m^3 / 3600s = (pi (0.200 x 10^-3)^2) v

v = 0.442 m/s

101325 + (1025 x 9.8 x 0.85) = P + (1025 x 0.442^2 /2 )

P =109763 Pa

through needle,

deltaP = 8 u L Q / pi R^4

= (8 x 0.0010)(0.2 x 10^-3 / 3600) (0.026) / (pi)(0.2 x 10^-3)^4

= 2299 Pa

P in vein = 109763 - 2299 = 107464 Pa

P_gague = 107464 - 101325

= 6139 Pa ......Ans

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