/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 69 An insulation company is examini... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 69

An insulation company is examining a new material for extruding into cavities. The experimental data is given below for the speed \(U\) of the upper plate, which is separated from a fixed lower plate by a 1 -mm-thick sample of the material, when a given shear stress is applied. Determine the type of material. If a replacement material with a minimum yield stress of \(250 \mathrm{Pa}\) is needed, what viscosity will the material need to have the same behavior as the current material at a shear stress of \(450 \mathrm{Pa} ?\) $$\begin{array}{ccccccccccc}\tau(P a) & 50 & 100 & 150 & 163 & 171 & 170 & 202 & 246 & 349 & 444 \\ U(\mathrm{m} / \mathrm{s}) & 0 & 0 & 0 & 0.005 & 0.01 & 0.025 & 0.05 & 0.1 & 0.2 & 0.3 \end{array}$$

Short Answer

Expert verified
The original material is a Bingham plastic with a viscosity of \(830 \, Pa.s\). The replacement material, to have the same behavior at a shear stress of \(450 \, Pa\), needs to have a viscosity of \(2000 \, Pa.s\).

Step by step solution

01

Identify the Material Type

The distinctive characteristic of the data provided is that material does not demonstrate flow (\( U = 0 \)) until a critical shear stress is reached (\( \tau = 163 \ Pa \)), which is the yield stress. This is typical of a Bingham plastic material. Hence, the type of material is a Bingham plastic.
02

Derive the viscosity of the current material

For a Bingham plastic, the equation of motion is \( \tau = \tau_y + \mu U \), where \( \tau_y \) is yield stress, \( \mu \) is viscosity, and \( U \) is speed. The speed above the yield stress increases linearly with increasing shear stress. Using the data where \( U = 0.1 \, m/s \) and \( \tau = 246 \, Pa \), subtract the yield stress to find the shear stress due to viscous flow: \( \tau - \tau_y = \mu U \) -> \( \mu = (\tau - \tau_y) / U \) -> \( \mu = (246 - 163) / 0.1 = 830 \, Pa.s \).
03

Determine the viscosity for the replacement material

In order for the replacement material to behave similarly to the current material at a shear stress of \(450 \, Pa\), it needs to have the same ratio of viscous to yield stress. The yield stress of this material is given as \(250 \, Pa\). We can use the same equation: \( \mu = (\tau - \tau_y) / U \) -> \( \mu = (450 - 250) / U \). Since we don't have the speed for the replacement material, we'll use the fact that the speed is proportional to the viscous stress, resulting in: \( \mu = 830 \, Pa.s \cdot (450 - 250) / (246 - 163) = 2000 \, Pa.s \).

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Most popular questions from this chapter

Tape is to be coated on both sides with glue by drawing it through a narrow gap. The tape is 0.015 in. thick and 1.00 in. wide. It is centered in the gap with a clearance of 0.012 in. on each side. The glue, of viscosity \(\mu=0.02\) slug \(/(\mathrm{ft} \cdot \mathrm{s})\) completely fills the space between the tape and gap. If the tape can withstand a maximum tensile force of 25 lbf, determine the maximum gap region through which it can be pulled at a speed of 3 ft/s.

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A concentric-cylinder viscometer is shown. Viscous torque is produced by the annular gap around the inner cylinder. Additional viscous torque is produced by the flat bottom of the inner cylinder as it rotates above the flat bottom of the stationary outer cylinder. Obtain an algebraic expression for the viscous torque due to flow in the annular gap of width \(a\). Obtain an algebraic expression for the viscous torque due to flow in the bottom clearance gap of height \(b\). Prepare a plot showing the ratio, \(b / a,\) required to hold the bottom torque to 1 percent or less of the annulus torque, versus the other geometric variables. What are the design implications? What modifications to the design can you recommend?
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