CBE 150A – Transport Spring Semester 2014

Laminar Flow Convective Heat Transfer

CBE 150A – Transport Spring Semester 2014

Goals:

By the end of today’s lecture, you should be able to:

obtain laminar flow heat transfer coefficients for: parallel plate exchangers

apply the applicable equations to determine: the exchanger outlet cup mixing temperature the required exchanger surface area

Laminar Flow Convective Heat Transfer

CBE 150A – Transport Spring Semester 2014

Outline:

I. Relevant equations

• Flat plate exchanger

II. Example – blood substitute heat exchanger

CBE 150A – Transport Spring Semester 2014

xy

- H

+ H

Uniform Plate Temperature TH

Temperature Profile (x,y)x = 0Fluid in laminar flow

Uniform Inlet Temperature T1

Laminar Flow Between Two Heated Parallel Plates

W

CBE 150A – Transport Spring Semester 2014

The Differential Equation (Basic Equations)

2

2

2

2

2

2

2

)(:

)(

2

:

123)(

yT

xTyuSimplify

xT

yT

xTyu

HUq

channeltheacrossvelocityaverageUWhere

HyUyu

w

CBE 150A – Transport Spring Semester 2014

The Differential Equation (Dimensionless Form)

Dimensionless variables:

H

H

TTTT

1 H

yy *1

2*

4PrRe

xH

UHxx

kC

WQUH

p

Pr

24Re

CBE 150A – Transport Spring Semester 2014

Subject to boundary conditions:** 01 yallforxat

*** 00 xallforyaty

010 ** xallforyat

2*

2

*2*1

23

yxy

Differential equation:

CBE 150A – Transport Spring Semester 2014

Yields:

0 0

**2

32exp

m n

nnm

mm yaxA

CBE 150A – Transport Spring Semester 2014

The solution yields T (temperature in the streamline) as a function of y (distance from the plate surface) but we probably don’t care !!

What we want is the average temperature in the exit fluid – the “cup mixing” temperature. So:

Cup Mixing Temperature

Temperature Profile (x,y)

H

H

mixingcup

dyHyU

dyyxHyU

0

2

2

0

123

,123

CBE 150A – Transport Spring Semester 2014

Cup Mixing Temperature

Temperature Profile (x,y)

*)89.1exp(91.0

3*2exp

0

2

x

ionapproximattermOne

xG

cm

m

mmcm

Equation12.5.17

CBE 150A – Transport Spring Semester 2014

Cup-mixing temperature along the axis of a parallel plate, laminar flow heat exchanger.

CBE 150A – Transport Spring Semester 2014

0

2

0

22

3/*2exp

3/*2exp38

4

mmm

mmmm

locloc

cm

Hcmloccmp

xG

xGNu

kHh

forsolutiontheingIncorporat

dxTThdTHUC

Local heat transfer coefficient

CBE 150A – Transport Spring Semester 2014

Local Nusselt numbers along the axis of a parallel plate laminar flow heat exchanger. Both surfaces at the same constant temperature. Also shown is the average Nusselt number plotted against x*, Dh = 4H

CBE 150A – Transport Spring Semester 2014

CFD Exercise – Blood substitute heat exchanger

In a surgical procedure on a kidney it is necessary to take the kidney “off-line” andnourish it with a blood substitute using an extracorporeal(outside the body) system. The blood substitute is passed through an oxygenator before it enters the organ, and it is necessary to raise the temperature of the liquid from 32 C to 38 C before returning it to the organ. A preliminary design under consideration has the following features. The blood substitute will flow through a parallel plate exchanger with a width (W) = 10 cm. and a channel height (2H) = 2 mm. The plates are maintained at TH = 40 C. The required capacity of the exchanger is a flowrate of 1 Liter /min

The liquid has the following properties:

Pr = 4.5 k = 0.6 W/ m K = 3 mPa s = 1250 kg / m3

What is the required length of the exchanger plate: