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Optimum control of heat exchangers with internal heat generation

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Abstract

This paper investigates the optimum control of a heat exchanger having internal heat sources from a reference steady state to a desired value. Both the wall and coolant are treated as distributed-parameter systems. Under certain constraints inherent in the operating conditions and physical limitations of the heat exchanger, the control function of the system, i.e. the heat generation rate which minimizes the deviation of the temperature distribution from an assigned pattern at a given time, is found through the use of a linear programming method. The effects of physical parameters on the optimal control function and the temperature response and distribution are examined. Experimental results are presented which compare favorably with the theoretical analysis. Heat exchangers to which these results apply include the electrical heater, a chemical reactor in which a chemical reaction occurs within the solid walls and a heterogeneous nuclear reactor.

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Abbreviations

A :

inside surface area of tube

A ij :

coefficient defined by (23)

a ij :

coefficient defined by (23)

b kj :

coefficient determined by (29)

C p :

specific heat of coolant

C pw :

specific heat of tube wall

C 0, C 1, ..., C n :

coefficients defined by (19)

d i :

inside diameter of tube

d o :

outside diameter of tube

d 0, d 1, ..., d m :

number defined by (22)

F kj :

function defined by (30)

G :

maximum allowable rate of wall temperature variation

g t (x, τ):

inverse Laplace transform of é t (x, s)

\(\bar g\) t (x, s):

transfer function defined by (10)

g Θ(x, τ):

inverse Laplace transform of \(\bar g\) Θ(x, s)

\(\bar g\) Θ(x, s):

transfer function defined by (11)

I(ϕ):

performance function defined by (5), (18), (24) or (25)

I 0 :

Bessel function of first kind, zeroth order

i :

an integer to identify axial location in heat exchanger

j :

an integer to identify time instant between zero time and final time

K :

ratio of surface heat conductance to heat capacity for coolant, = hA/ρC p V

K w :

ratio of surface heat conductance to heat capacity for wall, = hA/(ρC p V)w

L :

length of heat exchanger

M :

K/K w

m :

number of time intervals between zero time and final time

n :

number of space subdivisions which must be an even number

Q t :

a non-negative constant used to define the constraint for local coolant temperature

Q Θ :

a non-negative constant used to define the constraint for local wall temperature

s :

Laplace variable

T :

final time

t(x, τ):

deviation of coolant temperature (from initial steady state)

t*(x):

deviation of coolant temperature at final steady state

\(\bar t\left( {x,s} \right)\) :

Laplace transform of t(x, τ)

u :

flow velocity of coolant

V :

volume of coolant in heat exchanger, = πd 2i L/4

V w :

volume of tube wall of heat exchanger, = π(d 20 −d 2i )L/4

x :

axial distance measured from tube inlet

x i :

discreet distance measured from tube inlet as defined by (19)

y i :

non-negative auxiliary variable

y :

vector defined as {y 0, y 1, ..., y 4n+3m+6}

z i :

optimal control function as defined by (23)

z :

vector defined as {z 0, z 1, ..., z n }

z max :

= ϕ max/Φ

α :

coefficient matrix

β :

vector defined as {1, 1, ..., G}

Θ(x, τ):

deviation of wall temperature from initial steady state

Θ*(x):

deviation of wall temperature at final steady state

\(\bar \Theta \left( {x,s} \right)\) :

Laplace transform of Θ(x, τ)

ρ :

coolant density

ρ w :

wall density

τ :

time

τ j :

discreet time

τ*:

= τ−x/u

Φ :

volumetric heat generation rate at final steady state

ϕ(τ):

deviation of volumetric heat generation rate from initial steady state

ϕ max :

maximum allowable rate of volumetric heat generation

—:

Laplace transformed function

t :

coolant temperature

w:

wall

Θ :

wall temperature

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Authors and Affiliations

  1. Dept. of Mechanical Engineering, The Ann Arbor University, 48104, Michigan, U.S.A.

    H. S. Huang & W. J. Yang

Authors
  1. H. S. Huang
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  2. W. J. Yang
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Additional information

The work reported was supported by the Michigan Phoenix Project. Part of the paper was presented at the IFAC Symposium on Multivariable Control Systems, Duesseldorf, on October 7 and 8, 1968.

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Huang, H.S., Yang, W.J. Optimum control of heat exchangers with internal heat generation. Appl. Sci. Res. 23, 95–112 (1971). https://doi.org/10.1007/BF00413189

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  • DOI: https://doi.org/10.1007/BF00413189

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