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Assignment 6

The document contains 4 questions regarding heat and fluid flow problems. Question 1 involves steady heat conduction through a rectangular solid and nondimensionalizing the governing equations. Question 2 considers steady 2D channel flow driven by a moving top plate and neglecting the pressure gradient. Question 3 involves steady fully developed pressure driven flow between parallel plates. Question 4 deals with fluid flow between parallel plates due to a temperature difference, requiring nondimensionalization of the equations.

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17 views3 pages

Assignment 6

The document contains 4 questions regarding heat and fluid flow problems. Question 1 involves steady heat conduction through a rectangular solid and nondimensionalizing the governing equations. Question 2 considers steady 2D channel flow driven by a moving top plate and neglecting the pressure gradient. Question 3 involves steady fully developed pressure driven flow between parallel plates. Question 4 deals with fluid flow between parallel plates due to a temperature difference, requiring nondimensionalization of the equations.

Uploaded by

sourav ganguly
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Indian Institute of Technology, Madras

Department of Chemical Engineering


Assignment - 6
Date: April 09, 2024
Max mark: 20

Q1. Consider a heat conduction problem through a rectangular solid of


length 𝑎 and width 𝑏 as shown in the figure below.
𝑇 = 𝑇1

𝑇 = 𝑇0

𝑇 = 𝑇0 𝑏

𝑎 𝑇 = 𝑇0
𝜕2 𝑇 𝜕2 𝑇
The heat conduction equation is given as:
𝜕𝑥 2 + 𝜕𝑦 2
= 0. The boundary
conditions for the temperature has been shown in the above figure. Write
the dimensionless governing equations and the boundary conditions based
on the characteristic scales given below.
(a) 𝑥𝑐ℎ = 𝑦𝑐ℎ = 𝑎, 𝑇𝑐ℎ = 𝑇0
(b) 𝑥𝑐ℎ = 𝑎, 𝑦𝑐ℎ = 𝑏, 𝑇𝑐ℎ = 𝑇0
Q2. Consider the steady unidirectional flow of an incompressible
Newtonian fluid in a two-dimensional channel that is driven by an applied
pressure gradient (𝛿𝑝) and the motion of top plate (U). The problem setting
is same as that discussed in the lecture. Instead of pressure gradient
dominating the flow (which was worked out in the lecture), consider the
case of the moving top-plate that dominates the flow in the channel.
(a) Perform a scaling analysis, and non-dimensionalize the equations of
motion.
(b) Under what conditions, can the applied pressure gradient be neglected
in comparison with the motion of top plate? Express the condition in terms
of a suitable dimensionless number.
(c) Simplify the governing equations and boundary conditions. Solve the
simplified system of equations.

U0
y H
x

L
Q3. Consider a steady state fully developed pressure driven flow of viscous
Newtonian fluid between two infinitely wide parallel plates. Write the
simplified governing equations (continuity equation and momentum
balance in x and y directions) with appropriate boundary conditions. Choose
the following characteristics scales to make the equations dimensionless.
Simplify the equations based on order of their magnitude.

L
𝑢𝑐ℎ = 𝑢∞ , 𝑥𝑐ℎ = 𝐿, 𝑦𝑐ℎ = 𝛿(𝐵𝑜𝑢𝑛𝑑𝑎𝑟𝑦 𝐿𝑎𝑦𝑒𝑟 𝑇ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠)
Q4. Consider fluid flow between two parallel plates because of the
temperature difference between the plates. The temperature of the top plate
is 𝑇𝐶 and the bottom plate is 𝑇𝐻 as shown in the figure below.
Nondimensionalize the governing equation and boundary conditions. Write
the dimensionless equations in terms of dimensionless numbers. Also
explain the significance of those dimensionless numbers.
𝐿2 𝜈 𝜈 𝜇
(𝑎) 𝑥𝑐ℎ = 𝑦𝑐ℎ = 𝐿, 𝑡𝑐ℎ = , 𝑢𝑐ℎ = 𝑤𝑐ℎ = , 𝑝𝑐 = 𝜇 ∗ 2 , 𝑤ℎ𝑒𝑟𝑒 𝜈 =
𝜈 𝐿 𝐿 𝜌
𝑘 𝑘
( ) ( )
𝐿2 𝜌0 𝐶𝑝 𝜌0 𝐶𝑝
(b) 𝑥𝑐ℎ = 𝑦𝑐ℎ = 𝐿, 𝑡𝑐ℎ = 𝑘 , 𝑢𝑐ℎ = 𝑤𝑐ℎ = , 𝑝𝑐 = 𝜇 ∗
( ) 𝐿 𝐿2
𝜌0 𝐶𝑝

𝑘
Where = 𝐷𝑇 = Thermal diffusivity
𝜌𝐶𝑝
(𝑇−𝑇𝑐 )
Take dimensionless temperature 𝑇 ∗ =
(𝑇ℎ −𝑇𝑐 )

The dimensional governing equations are:


𝜕𝑢 𝜕𝑣
(a) + =0
𝜕𝑥 𝜕𝑦
𝜕𝑢 𝜕𝑢 𝜕𝑢 ∂p ∂2 u ∂2 u
(b) 𝜌0 ( +𝑢 +𝑤 )=− + μ( 2
+ )
𝜕𝑡 𝜕𝑥 𝜕𝑧 ∂x ∂x ∂z2
𝜕𝑤 𝜕𝑤 𝜕𝑤 ∂p ∂2 w ∂2 w
(c) 𝜌0 ( +𝑢 +𝑤 )=− + μ( + ) + 𝜌0 𝑔(1 − 𝛼 (𝑇 − 𝑇𝑐 ))
𝜕𝑡 𝜕𝑥 𝜕𝑧 ∂z ∂x2 ∂z2
𝜕𝑇 𝜕𝑇 𝜕𝑇 ∂2 T ∂2 T
(d) 𝜌0 𝐶𝑝 ( +𝑢 +𝑤 ) = k( + )
𝜕𝑡 𝜕𝑥 𝜕𝑧 ∂x2 ∂z2

𝑇1

𝑇0

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